llA'^ LIBRARY OF THK University of California. GIFT OF Class 2- Digitized by the Internet Arciiive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/e1ementsofalgebr00lillrich THE Elements of Algebra BY GEORGE LILLEY, Ph.D., LL.D. EX-PRESIDENT SOUTH DAKOTA AGRICULTURAL COLLBOB OF THE ^ UNIVERSITY SILVER, BURDETT & COMPANY New York . . . BOSTON . . . Chicago 1894 Copyright, 1892, By Silver, Burdett and Company. mniijcrsitg lirrss : John Wilson and Son, Cambridqk, U.S.A. PREFACE. Algebra is a means to be used in other mathematical work; it develops the mathematical language, and is the great mathematical instrument. If the student would be- come a mathematician, he must understand this language and possess facility in handling the various forms of literal expressions. Attention is called to the sequence of subjects as herein presented. Involution is introduced as an application of multiplication, evolution as an application of division, and logarithms as an application of exponents. Throughout the book the student is led to see that one subject follows as an application of another subject. The beginner is led to see at the outset that Algebra, like Arithmetic, treats of numbers. Algebraic terms and definitions are not introduced until the student is required to put them into actual use. Correct processes are cleariy set forth by carefully prepared solutions, the study of which leads the pupil to discover that method and theory follow directly from practice, and that methods are merely clear, definite, linguistic descriptions of correct processes. The book is sufficiently advanced for the best High Schools and Academies, and covers sufficient ground for admission to any American College. Great care has been given to the selection and arrangement of numerous examples and problems. These have been, for 18:5963 iv PREFACE. the most part, tested in the recitation-room, and are not so difficult as to discourage the beginner. It remains for the author to express his sincere thanks to W. H. Hatch, Superintendent of Schools, Moline, 111. ; to Professor W. C. Bojden, Sub-Master of the Boston Normal School, Boston, Mass. ; and to O. S. Cook, connected with the literary department of Messrs. Silver, Burdett & Co., for reading the manuscript and for valuable suggestions. GEORGE LILLEY. Pullman, Washington, June, 1892. PREFACE TO THE SECOND EDITION. In this edition the typographical errors have been cor- rected, and a page of examples has been added to Chapter XXVII ; also, the exercises have been carefully revised and corrected. Answers to the examples and problems have been prepared, and are bound in the book, or separately in flexi- ble cloth covers. The answer-book is furnished for the use of the class only on application of teachers to the publishers for it. The publishers and the author desire to express their appreciation of the very favorable reception accorded to the first edition. September, 1894. CONTENTS. CHArrEK PACK I. First Principles l II. Algebraic Addition 19 III. Algebraic Subtraction 27 IV. Algebraic Multiplication 35 V. Involution 52 VI. Algebraic Division 60 VII. Evolution 79 VIII. Use of Algebraic Symbols 99 IX. Simple Equations 104 X. Problems Leading to Simple Equations 109 XI. Factoring 119 XII. Highest Common Factor 141 XIII. Lowest Common Multiple 155 XIV. Algebraic Fractions '164 XV. Fractional Equations •. . . 201 XVI. Simultaneous Simple Equations 215 XVII. Problems Leading to Simultaneous Equations . . 238 XVIIL Exponents 248 XIX. Radical Expressions 263 XX. Logarithms 296 XXI. Quadratic Equations 312 vi CONTENTS. CHAPTEK PAGE XXII. Equations which may be Solved as Quadratics . 330 Theory of Quadratic Equations 339 XXIII. Simultaneous Quadratic Equations 345 XXIV. Indeterminate Equations 355 XXV, Inequalities 363 XXVI. Series 373 Arithmetical. 373 Geometrical 379 Harmonical 384 XXVII. Ratio and Proportion 388 APPENDIX 401 INDEX TO DEFINITIONS. PAOI Algebra 118 Binomial 46 Coefficient 20 Equation, Biquadratic . 334 " Degree of, Roots of 35, 108 " Exponential 309 Literal 206 " Syminetriciil 347 Expression, Algebraic 90 " Compound 23 ** Homogeneous 349 ** Imaginary 286 ** Irrational 263 ** Mixed 164 " Simple 21 Factor 119 Figures, Subscript 227 Fraction, Complex 188 Continued 190 Identities 104 Index 79 Mean, Arithmetical 378 " Geometrical 383 " Harmonical 385 Nlonoraial 21 Multiple ,155 " Common 155 Multiplication, Algebraic 38 Vlll INDEX TO DEFINITIONS. PAGE Numbers, Algebraic, Absolute ... 14 " Known 107 " Negative 11 " Scale of 12 " Unknown 107 Polynomial 23 Power 35 Progression, Arithmetical 373 " Geometrical 379 " Harmonical 384 Quantity <.....• 389 Keciprocal ^8 Roots 79, 340 Signs, Algebraic 13 " Double 80 " Law of . 38, 61 " Radical 79 Subtraction ' 34 Surd, Similar 263 " Entire, Mixed 264 " Quadratic 290 Symbols of Abbreviation 7 " of Aggregation, of Relation 6 " of Operation 1, 99 Terms 3, 90 " Like 20 Term, Absolute 207 " Degree of. Dimension of 345 Value, Absolute • < 14 ** Numerical 9 ELEMENTS OF ALGEBRA CHAPTER I. FIRST PRINCIPLES. 1. In Algebra figures and letters are used to represent numbers, instead of figures, as in Arithmetic. Thus, we may use x to represent the number of dollars in a man's business, the number of cents in the cost of an article, the number of miles from one place to another, the number of persons in our class, etc. In Algebra, the letter x is reasoned about and operated upon just the same aa the numbers which it represents are reasoned about and operated upon in Arithmetic. 2. Symbols of Operation. The signs +, — , X, and -h, are used to deuute the algebraic operations addition, sulv traction, multiplication, and division, that in Arithmetic can actually be performed. + is read 'plus; — is read miniis ; X is read multiplied by; H- is read divided by. A dot or point is sometimes used instead of the sign X. Thus, a X 6 and a • h both mean that a is to be multiplied by h. The multiplicand is usually written before the multiplier. Dimsio-ii in Algebra is more frequently represented by placing the dividend as the numerator, and the divisor as 2 ELEMENTS OF ALGEBRA. the denominator of a fraction. Thus, a -i- b, or - , means b that a is to be divided by b. Eead a divided by b. m Note. Do not read such expressions as — , m over n; it is meaningless. 3. We must be careful to distinguish between arith- metical and algebraic operations. The former can actually be performed, whereas many operations in Algebra can only be indicated. Thus, suppose a man owes ^ 5 for a vest and % 20 for a coat, actual addition gives $25 as his total indebtedness. But if the number of dollars he owes for the vest be represented by m, and the number of dollars that he owes for the coat be represented by n, his entire debt can only be indicated. In order to show that the number represented by m is to be added to the number represented by n, we use the sign + written between them ; thus, m + n. Exercise I. Eead the following algebraic expressions : 1. 0! + 100 ; a + 10 - 2 ; & - 2 ; & - 100 + 8. 2. a -\-b; m -\- n -{- ^ \ m + s ~ r; a — b + m. 3. c + 2x5; c -10 X2; s-nX r-20. 4. q + t + S X m; c + m^n — s - q; — - + c-^a—p -{- 1 - x. a Indicate by means of algebraic expressions the following: 5. The sum of m and n. The difference between m and n. The sum of x, y, and a. 6. The sum of m, n, and r diminished by t. If you had m cents, earned n cents, and are given r cents, and then spend t cents ; how many cents will you have left ? FIRST FRINCIPLKS. 3 7. John has m apples, Henry has n apples, and Charles has b apples ; express the number of their apples. How many more have John and Henry than Charles ? 8. If you buy goods for a dollars and sell them at a gain of b dollars, express the selling price. 9. I buy goods for m dollars and sell them at a loss of 71 dollars ; express my selling price. 10. Henry had x marbles ; he gave John vi marbles, and Charles ii marbles. How many had he left ? 11. I pay n cents for a reader, x cents for a history, y cents for a grammar, 6 cents for car-fare, and have m cents left ; express the number of cents that I had at first. 12. A boy earned a dollars, then received m dollars from his father, n dollars from his mother ; and spent k dollars of what he had for books, x dollars for a coat, and y dollars for a sled. Express the number of dollars he had left. 4. The Sign of Multiplication is generally omitted in Algebra, except between figures. Thus, bah means bXaXb, prstuz means p X r X s X t X v X z ; J • 3 • 4 • 5 means 2 X 3 X 4 X 5, or 120. Again, if the numher of gallons in a cisk of cider is represented by a, and the number of cents in the cost of one gallon is represented by m, then the number of cents in the cost of n casks is represented by amn. 5. In the expression 5 + 2«o — a+ -— — : 5, m 2 am ^ 2 ab, a, — , and -ttt- &re called Terms. n ooc ELEMENTS OF ALGEBRA. Exercise 2. Eead and state the meaning of the following algebraic expressions : 1. 5ahx-\ ah. Result : 5 times a times h times c X, plus 7n times n divided by c, minus a times b ; etc. 2. kl + 1-t; PQrs + ab cd + mnx y — 80. en b , ^ ^ ^klx abed a. i . i-i 3. amnpgr — cdXo-\-- ; Zb d+ 11— r. , bwyz mnop 4 bx'^+12pqrst-Q^hk-^a-\-imz. ab „ S ab d — 10 mnr + Imnr st 5. . ad — 1 6. h + '^-^+u; 2^+2^^+^^. ?±fZ^* + „ + *_±f . 4 y u X a I 6. It is customary to write the letters in the order of the alphabet. In a product represented by several letters and numbers, the num- bers are written first. Thus, cX&XaX5X3 is written 3 X 5 ahc ; both mean 15 a 6 c. Also, s X r X n X m X 25 is written 25 mnr s. Exercise 3. Write algebraic expressions for the following : 1. The product of x, y, and z ; of m, n, and 5 ; of 3 and xy, of 5, a, b, and S X mn, 2. The product of a and b divided by their sum. Their product divided by their difference. FIRST PHINCIPLES. 5 3. The product of m, n, r, and 25 divided by the sura of m and n. The same product divided by the difference of VI and 71. 4. A travels at the rate of 3 miles an hour ; how many hours will it take him to travel 30 miles ? How many hours to travel a miles ? To travel m n miles ? To travel 60 a in n miles ? 5. A man bought 18 loads of wheat, of m bushels each, at n cents a bushel ; how many cents in the entire cost ? 6. In example 5, suppose that he sold the wheat at a gain of r cents a bushel ; how many cents did he gain ? How many cents in the selling price ? 7. In example 5, suppose that he sold the wheat at a loss of a cents a bushel ; how many cents would he lose ? how many cents in the selling price ? 8. A man bought a boxes of peaches, each containing h peaches, at c cents a peach ; and m baskets of grapes, each containing n pounds, at r cents a pound. How many cents did he pay for both ? 9. A man worked n hours a day for m days, at a cents an hour. With the money he bought a coat for x cents ; how many cents had he left? 10. One boy sold a apples at c cents each ; another sold n peaches at m cents each ; a third sold r peai-s at t cents each. How many cents did they all receive ? 11. I buy 5 tons of coal at SIO per ton, and pay for it in cloth at S2 per yard ; how many yards will it take? I buy a tons of coal at h dollars per ton, and pay for it in cloth at m dollars a yard ; how many yards will it take ? 6 ELEMENTS OF ALGEBRA. 12. A man works n weeks at h dollars a week, and his son works m weeks at r dollars a week. With the money they pay for c cords of wood at d dollars a cord ; how many dollars have they left ? 13. If 5 cords of wood cost $15, how many dollars will 3 cords cost? If c cords cost $m, how many dollars will n cords cost ? 14. A man drove 3 hours at the rate of 10 miles an hour ; how many hours will it take him to walk back at the rate of 6 miles an hour ? If he drives 3 days n hours each day, at the rate of t miles an liour, and 5 days m hours each day, at the rate of s miles an hour, how many hours will it take him to return over the same distance, at the rate of r miles an hour ? 15. If you buy t tons of coal at the rate oi %d for n tons, and sell it at a loss of $ Z on each ton, how many dollars will you receive ? Suppose you sell at a gain of %h on each ton, how many dollars will you get for it ? Suppose you sell all of it for r dollars, and make a profit, how many dollars profit will you get ? 7. Symbols of Relation. The signs =, >, and <, are used for the words, equals, is greater than, and is less than, respectively. Symbols of Aggregation. The signs ( ) , [ ] , { } , and , are used to show that the terms enclosed by them are to be treated ns one number. They are called parenthesis, bracket, hracc, and vinculum, respectively. Thus, (2 ft + 6) (3 X - y), [2 a + h][3x - yl {2 a + b] {3 x ~ i/], 2a i- b X 3x — y, each shows that the number obtained })y adding the terms 2 a and b is to be multiplied by the result obtained by subtracting y from 3 x. FIRST PRINCIPLES. 7 Sjrmbols of Abbreviation. The signs (of deduction) .*., (of reason) •.*, and (of continuation) ...., are used for the words, heiice or therefore, since or hecmise, and so on, respectively. 8. Since 81 = 9 X 9, or written 9^ for brevity, 81 is called the second power of 9. Since 27 = 3 X 3 X 3, or written 3^ for brevity, 27 is called the third power of 3. Similarly a^, {m n)\ (m + n)^, are called second poioers of a, m n, and m -^ n\ also a^, (m n)^, {m -\- n)^, are third powers of a, m n, and m -\- n. cfi means a X a ; a^ means a X a X a \ etc. In general, ft" is called the nth power of a, read a 7ith power. 9. In the expression a^ + h^c^ — 3 x"; 2, 4, 5, and ?i are called Exponents, h^ c^ means bxbxbxbxcxc xcX cXc; 4 and 5 are used for convenience to show how many times b and c are used as factors. We must be careful to keep in mind the meaning of each indicated operation when rea(hng an algebraic expression. Thus, the expres- sion 5 X* j/"^ — 2 a* 6 («' — 6*)^ + 3a^c*d"* means, five times the third power of X times, the second power of y^ minus two tim^s the fourth power of a times b times the fifth power of tlie expression in the parenthesis, a seventh power minus b sixth power, phis three times the fifth power of a times the fourth power of c times the with power of d. Exercise 4. Read the following : 1. m^;3m^x^;5m^nY,'fh(M^(^iab^-hh,vL^-n^\ lOaVr^. 2. m^n^ -f 5 a^bxi/ - 3 m^^ x^; m^n^ -'lab m n + (H'K 3. 10 (ft 6)^0; (?n3n8) (m nf\ (a^ - „)2. (,„2 _ 3 ^^^a 4. (m n - m3)3; 3 (v^ b {<i - }?'^- (n^ + />") (a^ - b^f- 8 ELEMENTS OF ALGEBRA. 5. 3 oTlf', 3 {a bf ; a^ (b^ - c^ - d'^f^ (m^ n^) (m n)\ 6. (10 m + n"^) (10 n^ - m^f < 15 a (x - i/)^ (x + yf ; (1^ + 1)^ + d^ + ^7 (c5 + d^y b{a + h + ef ^ ^'' ^ c2 + 6^3 ' 7. •.•a + 2:c=:6 + ^, .'.x = h — a\ {a'^ — c''f={7n?+n^f, ... a"* - C'* =r m2 ^ ^2 . ^2 - ^-3 = 2 ^3 _ 2 ,^,2^ .-.0.^3 = ,^,^2 . x-\-x-^x-\'X-\- to n terms =^ nx\ a X a X a X a X .... to 71 factoi'S = a" ; 1 -^ x ■\- x^ + st? -\- .... ..2 I ^, ^^ I f^ ., f^,„«o _ ^('''" ~ ^) 1 -a? a + ar + a ?'^ + a ?^ + . . . . to 7«. terms = , . r — 1 Write algebraic expressions for the following : 8. The sum of m and n. The double of x. The second power of the sum of a and h. The second power of differ- ence between x and y. Five times the third power of the difference of x and y. 9. The second power of the sum of x second power and y. The second power of the sum of x and y second power. The product of the fourth power of x, the third power of ?/, and the second power of 7n. The product of the first power of x and three times the nth power of y. The product of x second power plus y second power, and X second power minus y second power. 10. The product of the sum of x second power and y by n a. Five n third power minus seven m n plus six a second power, m third power minus two times b second power c plus n fourth power is equal to n times y. 11. Seven times m fourth power times n second power minus two times -^t seventh power times m third power plus three times a tliird power times h second power plus eight times a second power times b third power plus five FIRST PRINCIPLES. 9 times a fifth power. Since a plus h equals m minus w, therefore the second power of a plus h is equal to the second power of m minus ii. 12. Therefore, x is equal to m third power, because x plus three m third power is equal to two x plus two m third power, a plus a plus a, and so on to n minus two terms, equals n minus two times a. The second power of m plus yi, divided by m minus n is less or greater than m times a plus h plus c plus o? plus e. a less than 6 is equal to iii greater than n. 13. A horse eats a bushels and an ox h bushels of oats in a week ; how many bushels will they together eat in n weeks ? If a man was a years old 50 years ago, how old will he be x years hence ? 10. The Nomerical Value of an algebraic expression is the number of positive or negative units it contains, and is found by giving a particular value to each letter, and then peiforming the operations indicated. Thus, If a - 3, 6 = 4, x = 5, y = 6, find the numerical values of : ,2,. 9 6x« 25 a* 1/2 Replacing? the letters in each expression hy the •particular values given for them, we have Process. 4 a«6« = 4 x 3^ X 4» = 4X9X64 = 2304. 9h3* 9 X 4 X 5« 25rt»y2~25 X 3»X 62 9 X 4 X 125 25 X 27 X 36 = A 27' U. If one factor of a product is equal to 0, the whole product nmst be equal to 0, wlmtever values the other factors 10 ELEMENTS OF ALGEBRA. , may have ; and it is also clear that no product can be zero unless one of the factors is zero. Thus, ah is zero if a is zero, or if b is zero; and if ab is zero, either a or 6 is zero. Again, if a; = 0, then a^h'^xy^ = 0, also ax{y^ + 62; + «^) == 0, whatever be the values of a, 7;, 3/, and z. Exercise 5. If <x = 6, & = 2, c = 1, a; = 5, ?/ — 4, find the numerical values of the following algebraic expressions : L 3c2; 72/3; 5a&; 9^?/, 8Z?3. 3-^5. ^^8.7^4. ^^^.10. 3 ^^4^ 2. 9 6*; 2 a a; ; 3/^ , 10 x^ ; \y^\ 5 6?/; |^ r^^ ; ^^ab cxy. 3. 3^2 ^^ 5 1^ ^ ^ ; 7 c^ ; f .«3 ; a* x^ ; 8 a^^ 2 3/* ; | acxy. li 7n = 2, n = 3, p = I, q = 0, r = 4, s = 6, find the values of: 4 ^^. Pi,„,2^. 4^^^■^ ?ZL!!_^ ^J!^. o«»9n. 2m2g p; 8^'^^ r, 6 06 5 s 4 5 7776 r- 27 m'- 64 2' 9 7?t3 ' 6 ' ' /^ ' 54 ^m ' 32 ' r» ' 3" Example 6. Find the value of 5 6^ + j% x y - 5 a^ - ^a^b^, when (1 = 2, 6 = 3, x = 5, and i/ = 10. Replacing the letters in the expression by the particular values given for them, we have Process. 563 4- ^3_xy _ 5a2_ I Q^'2^3 = 5 X 33 +^3_ X 5 X 10 - 5 X 22 - I X 22 X 33 = 5X27 + 3X5 -5X4-3x27 = 49. Example 7. Find the value of mny^-\- mhi xyt + m"nh's — -j-^ , wlien ?7i = 5, u = 2, r = 3, s = 4, X = 0, and y = I. FIRST PRINCIPLES. 11 Process. mny*-Hm*nxytfm%V8-^ = 5 X 2 X IHO + S-'X 2» X 3<X4- ^ ^ ^.^ 25 X« = 5X2X1 + 25X8X81X4-——^ = 10 4- 64800 - 50 = 64760. If a = 1, b = 2, c = 3, and d = 0, find the numerical values of the following algebraic expressions : 8. 10 a — 4:b -{- 6 c -\- 5 d] ab + hc + ac — da, 9. 6 ab-Scd-{- 10 a d-2b c+ 2bd; 2 be + 10 cd. 10. a^ + Ir^-\-c^-(P; abc + 10 bed + 5 ac d + 'S abd. .11. a* + h^ -\- c'-d; +ob-8e -{- ad; —40ad-\-ab. 12. 5a+3c-G6+6(/; 36co?-|-2acc?-10rt&rf. 13. 5/>r3 + .^3 ^_ /,3_ i.25a&3c; 15 «2 ^_ jj. r* + 10 a &. .Cw/2/,4 .IftS 14. c3- 8 «(/6^- 5 ftio^. j^^y. c3 ai« 62 c ' 15. 125a6cc/*m + ^,-^^l^'; «» 4. ?,3 + ,3 + ,^;^. 80 X V\ 2 ., ?,3 4. iL ^ 2;{c - ^^, 3 «2i3,;6^ ^ '_' _ 2^' /; a' 8 , , 3 ) a ** 2^-'^ c^ abc 8 ^/ /•" 8<* ^'^ r^ 19. J a f^ ;/ -\- I n — | ^^2 d^xy\ Wa^^^^ '^ ~q~ ~ ^- o o 12. Negative Numbers, if a person owes a debt of ten dollars, and \\di» but .six dollars in money, he can pay the debt only in part. 12 ELEMENTS OF ALGEBRA. For his six dollars in money will cancel only six dollars of his debt, and leave him still owing four dollars; we may consider him as be- ing worth four dollars less than nothing. The total number of dol- lars that he is worth may be represented by — 4, because it will take four dollars in addition to the six dollars to pay the debt. If a person gains eight dollars and loses eleven dollars, the number of dollars in his net loss may be represented by — 3, because it will take three dol- lars in addition to the gain to balance the loss. Similarly, if he gains 100 dollars and loses 120 dollars, the number of dollars in his net loss may be represented by — 20. To enable us to represent these num- bers, it is necessary to assume a new series of numbers, beginning at zero and descending in value from zero by the repetitions of the unit, precisely as the natural series ascends from zero. To each of these numbers the sign — is prefixed. The negative series of numbers is written thus : 10, -9, -8, -7, 6, -1, 0. For convenience the algebraic series of numbers is represented as follows : Scale of Numbers. We may conceive algebraic numbers as measuring distances from a fixed point on a straight line, extending indefinitely in both directions, the distances to the right being positive, and the distances to the left nega- tive. From any point on the line, measuring tovjard the right is positive and tovmrd the left negative. -^-h -lis -ilo -l5 llO + 115 In the above illustration consider A the zero or starting-point on the scale of numbers, and the distance between any two consecutive numbers one unit. The distances to the right and left of A are posi- ^ i FIRST PHINCIPLES. 13 tive (+) and negative (— ), respectively, as indicated by the direc- tions of the arrows. To add + 9 to + 4 (read 9 and 4 in tlie positive series) ^ we start at 4 in the positive series, count nine units in the positive direction, and arrive at 13 in the positive series. That is, + 4 + (+ 9) = 13. To add + 9 to — 4 (i-ead 9 in the positive series and 4 in the negative series) f we begin at 4 in the Jiegative series^ count nine units in the positive directioiiy and arrive at 5 in the positive series. That is, -4-f (+9) = -+-5. To add — 9 to -I- 4, we start at 4 in the positive series, count nine units in the negative direction, and arrive at 5 in the negative series. That is, + 4 + (- 9) = - 5^ To add — 9 to — 4, we start at 4 in the negative series, count nvie units toward the left, and arrive at 13 in the negative series. That is, - 4 + (- 9) = - 13. To subtract + 9 from + 4, we start at 4 in the positive series, count nine units in the negative direction, and arrive at 5 in the negative series. That is, + 4 - (+ 9) = - 5. To subtract + 9 from — 4, we begin at 4 in the negative series, count nine units in the negative direction, and arrive at 13 in the negative series. That is — 4 — (4- 9) = — 13. To subtract — 9 from + 4, we begin at 4 in the positive series, count nine units in the positive direction, and arrive at 13 in the posi- tive series. That is, + 4 - (- 9) = + 13. To subtract —9 from —4, we start at 4 in the negative series, count nine units in the positive direction, and arrive at 5 in the positive series. That is, - 4 - (- 9) = -|- 5. 13. The sign + is often omitted before a number in the positive series. Thus, the numbers 3, 5, and 6, taken alone, mean the same ^ (+3), (-f- 5), and (+ 6), showing that the numbers are in the positive series The sign — must always be written when a number is in the negative series. Thu.s, the numbers 3, 5, and 6, taken in the negative series, are written (- 3), ( - 5), and (- 6). The Algebraic Signs -f- and — mark the direction that the numbers following them are to take. These signs are 14 ELEMENTS OF ALGEBRA. used to indicate opposition (opposite direction), also opera- tion. The former is called the positive, and the latter the negative sign. An Algebraic Number is one -Which is represented by an algebraic term vnth its sign of direction. Thus, + 3,-3, — a, and + 5 a are algebraic numbers. Absolute Value shows what place a number has in the positive or negative series. Thus, + 3 and — 3 have the same absolute value ; that is, three units. Absolute Numbers are those not affected by the signs -f or —. Example. The meaning of an algebraic expression, as 3a;2+(-2a6)-[c-(-2/)], is explained thus : To 3 x^ units in the positive series add 2ab units in the negative series, and from their sum subtract the expression in brackets, c in the positive series minus y in the negative series. The signs written before the terms (—2 a 6), (— y), and before the bracket, indicate operation. The sign written before 2 ah and y, also the sign under- stood before 3 x^ and c, indicate opposition. Exercise 6. 1. Over how many units and in what series of numbers would a point move in passing from 4-3 to — 8? — 10 to + 1? +5 to +15? -12to-l? -lto-12? 15 to 5? 9 to 9 ? — 5 to — 5 ? 2. Which is the greater, or — 6 ? 3 or — 3 ? — 5 or — 3 ? + 10 or — 1 ? + 50 or — 50, and how many units ? 3. How many units is + 6 greater than 0, + 3, — 3, — 6, and — 5 ? How many units is — 5 less than 5 ? How many units is a less than b ? FIRST PRINCIPLES. 15 4 If a point start at (-f 3) and move three units to the right, then five units to the left, where is it? Express its distance from 0. 5. Suppose a point start at (+ 2), and move six units to the right, then eleven units to the left, where is it ? 6. Where is the point which, starting at (— 5), moved (— 3), then (+ 8) ? Express its distance from the starting- point. 7. Suppose a point starting at + 3, move + 2, then — 7, then -f 5, then — 6, then + 10, then — 1-1, where would it be ? Express its distance from -I- 3. Explain the meaning of: 8. 2 [3 6 -f (- 5 a)] - 5 [(- a) + (+ h)]. 9. (+ 8a;) 4- [+ 3x-(+ 12 y) + (- x) - (8 y)]. Also the meaning of the signs + (as used or understood) and — . Explain the meaning of: 10. 6«5A2 -f (-\-a^}fi) -f (^aH^c^) + (-aH'^) + (+aH^) + 20 a2^c2 + (- aH^) + (- a^b^). 11. aU^ ^ ^_ ^^.10) ^ (^ ^6 ^5^ ^_ (_ y a.) + ,;i3 ,^ft - (4- a^° m^). 12. + (+ a^) ~{-\-h^)- (+ a^) - (+ ^). 13. (x + 7/)2 + (a + xf - (x + 7/)2 - (a + x)^ Find their numerical values when a^I?^c^m = n = k = y = x=l. 16 ELEMENTS OF ALGEBRA. Eead the following expressions, and find their numeri- cal values when a = 0, b = 1, c = 2, d = S, e = 4, n = 6, and m = 6 : 14. c5 - (+c) - (+ n^) (+ c) - (+ d^) + a^(+ h^) (+ <^\ 15. 3 [g + (+ 7ij\ - 5 (+ c) -h (a + 6) + 2 (+ «) ^ 6. 16. (+c) [a + (4- ?i) + (+^) - (+6)] - (+ m) -r (+ d), 17. [(+m) - (+rO + (+m) - (+e)(+c)] - [(+m) (+6)]. 18. d^^{^d')^'2{+h')-^{+c')-{+c^)-^d-e^-^{-\-4:% If tt = 5, & = 4, c = 3, t^ = 2, and e = 1, find the numeri- cal values of the following expressions : 19. (+a2)4-(+/^3)_(+c2)--(+e5); a'^c-ahl?+d^-^{a'^W'). If a = 1, 5 == 2, c = 3, c? = 4, c = 5, find the values of: 26+2 3c-9 e2-l d^ 8a24.3?>2 4,2^552 c^^.^ e-3 6-2"^e + 3' F' «2+Zy2 + c2-62 ^2 d>-h'' eh c ' b'^+d^-bd' e^+ed+d^' a^ + 4a%+6a%'^ + 4:ab^-{ -h\ 28 12 a^ + Sa^b + Sal)^ + b^ ' o? + h^j\-c^^ d?-c^-}p^' 24. aJ— 156-^5; — ^ ^ ; . a -\- b c + ft o + c a*-4a3(?4-6«2c2_4r?.^3_|_^4 ,^ , ,^ ^, ^^ 25. ,4 ^xq . ...22 .z.q 7 4 ; 12e-4a-^(2aX5)-26. b*—4:Wc+bb^c^—4:b(f-\-c^ ^ ^ 26. [(12e-4a)-^2«] XZ^; a^ ^ {aHU^) ^ (^c^ - a'^y 29. FIRST PRINCIPLES. 17 27. -T-i ;, , > + ^ . . , , ; (« + ^) (6 + c). 28. (6 + c) (c + rf) + (c + rf) (rf + e) + -^_^ji^_^. 30. 66-^-(a — c) — 3rf + a6ctZ-i-24a;c + 5f]^-Ho + a X e. Express the following statements in algebraic symbols : 31. To the double of a add 6. 32. To five times x add h diminished by one. 33. Increase h by the sum of a and x divided by y. 34. Write x, a times. What is the sum oi x-\- x-^x ■\- .... written a times ? 35. Write three consecutive numbers of which n is the least. 36. Write five consecutive numbers of which m is the greatest 37. Write w, a minus 1 times ; also m plus n times. 38. Write seven consecutive numbers of which x is the middle one. 39. Write a, icth power, minus y, nth power. 40. To the double of x, increased by a divided by &, add the product of a, J, and c. 41. To the product of a and h add the quotient of x di- vided by a, and divide their sum by y diminished by c. 2 18 ELEMENTS OF ALGEBRA. 42. Write a exponent n plus the quotient of x divided by y, minus h times the quotient of h divided by the ex- pression, a exponent c plus & exponent m, is greater than h minus x, 43. Write x fifth power minus h sixth power ])lus y to the ??ith power, divided by z to the n\\\ power, is less than g- tenth power. 44. Write a to the ??-th power divided by h exponent m, minus x exponent n, equals a minus h, divided by the sum of a second power and h third power. 45. Write c fourth power divided by a second power, minus the product of ^ and y, plus x .... written n times, equals a exponent m. 46. X exponent m, plus the fraction, a fifth power minus three times a second power h third power, divided by x minus y, equals x minus y, added to the sum of 4 a and h minus m, plus 1 divided by x to the nth power. 47. Five times the third power of a, diminished by three times the third power of a times the third power of &, and increased by two times the second power of h. 48. Three times x exponent 2, minus twice the product of X exponent 3 and y, plus the third power of a. 49. Six times the third power of x multiplied by the second power of y, minus a exponent 2 times the fourth power of h. 50. a times the second power of n, divided by x minus ?/, increased by six a times the expression x plus y minus z. ALGEBKAIU ADDllluN. 19 CHAPTER ir. ALGEBRAIC ADDITION. 14. In Art. 12 it was shown that to add a positive miniber means to count so many units in the positive di- rection, and to add a negative number means to count so many units in the negative direction. In Algebraic Addition of several numbers, we count from the phice in the series occupied by any one of the num- bers, as many units as are equal to the absolute value of the numbers to be added and in the direction indicated by their signs. Thus, ExAMPLK I. Fiml the sum of 3 a and — 9a Solution. 3 a signifies a taken 3 times in the positive series, and — 9a aigniiies a taken 9 times in the negative series We count from + 3 a, 9 a units in the negative direction, and a is tiikeii ia all 6 times in the negative serie.*?, or —6a. That is, 3a+ (— 9a) = - 6a. Similarly (-}- 9 a) + (- 3 a) = 4- 6 a. Example 2. Find the sum of a, 2 6, and (— 3 c). > + "N^ >i +Z^ A D c o -3C < 20 ELEMENTS OF ALGEBRA. Explanation. Suppose these algebraic numbers to be accurately measured as represented on the line of numbers A C. Start at By then count 2 b units in the positive direction and arrive at G. Now count 3 c units in the negative direction, and arrive at D in the posi- tive series. Thus, (+ a) + (+ 2 6), or a + 2 6 = ^ C ; a -{- 2b + (- 3 c), or a + 2h - 3 c = A D. The sum of the algebraic numbers is equal in absolute value to A D in the positive series. That is, (+ «) + (+ 2 6) + (- 3 c) = a + 2 6 - 3 c. Hence, The sum of several algebraic numbers is expressed by con- necting them loith their proper signs. Notes 1. The sum of several algebraic numbers is the excess of the num- bers in the positive series over those in the negative series, or the excess of the numbers in the negative series over those in the positive series, according as the one or the other has the greater absolute value. Thus, in Example 1 the algebraic sums are —Qa and + %a. In Example 2 the algebraic sum is A D in the positive series. 2. The sum of algebraic numbers is the simplest expression of their aggre- gate values. 3. Algebraic addition is not always augmentation as in arithmetic. Thus, (+ 7) + (- 5) = 2 ; also (+ 8) + ( -ll) = - 4. 15. A Coefficient of a term is d^ factor showing how many times the remainder of the term is taken. Thus, In the term 5 abm, 5 is the coefficient of a b m, and shows that a 6 m^s taken 5 times ; 5 a is the coefficient of 6 m ; 5 ab is the co- efficient of m. In the term 4 m (a 6 - 2 a), 4 is the coefficient of m (rt 6 — 2 a) ; 4 w is the coefficient of (a 6 — 2 a). Note. A coefficient may be numerical or literal. When no nnmerical coefficient is expressed, 1 is always understood to be the coefficient; as, x ; xy°. Like Terms are those having the same letters affected with the same exponents. Thus, ALGEBRAIC ADDITION. 21 2m^ujc^ </*, m"^ n jfi y*, and — 10 m* n x* y^ are like terms, as are also bx^^y^z* and — 3x"y*z*; but dx*y^ and 5x*i/*2* are un- like terms. Like terms are said to be similar. A Monomial or Simple Expression consists of one term; as, x: lU (( l>c: — 5 a^ j^. 16. To Add Similar Monomials. I. When all the Terms are Positive or Negative. Add the numerical coefficients; to the sum, annex the common symbols^ and prefix the common sign. II. When Some of the Terms are Positive and Some are Negative. Add separately th£ numerical coefficients of all tlie positive terms and the numerical coefficients of all the negative terms; to the difference of these two results^ annex the common symbols, and prefix the sign of tlie greater sum. EXAMF'LE 1. Find the sum of 10 a; y*, - Sxy^ 4x?/*, and —\lxy^. Explanation. For convenience write the terms as shown in tlie margin. The sum of the coeffi- cients of the positive terms is 14, and the sum of the coefficients of the negative terms is 'SI. The difference of these is 17, and the sign of the greater sum is negative. Hence, the required sum is — nxy*. Uxf, Process. + Axy^ - 3xy^ - Uxy^ — M xy^ — 17x2/* Example 2. Find the sum of (x -f y), 1.1 (ic + y), - 2.9 (x + y), .29 (x + J/) . - i (x + y), and 1 .26 (x-\-y). Explanation, (x+y), enclosed in parentheses, is treatt^l as a simple symbol. The coefficients of (x + 1/) are 1, l.l, 2.9, 29, i, and 1.26. The sum of the coefficients of the positive terms is 3.65, and the sum of the coefficients of the negative terms is 3.15. The difTerence of these is .5, and the sign of the greater sum is positive. Etc. Process + C-^ + y) 1.1 (x + y) + .29(x-f-2/) -f 1.26 (x 4- 2/) -2.9 (x + y) - H^ + y) + .5(x-fy) 22 ELEMENTS OF ALGEBRA. Exercise 7. Find the sum of: 1. (+ 2 a), (+ a), (+ 4a), (+ 3a), (+5 a), and (+ la). 2. (+ 5 a a?), (+ 2 a x), (+ 6 a x), (+ a x'), and (+ a x). 3. (+ 6 c), (+ 8 c), (+ 2 c), (+ 15 c), (+ 9 c), and (+ c). 4. (-6a&c), (+4«&c), i+ahc), (-2abc), and (+5a&c). 5. (-fa;2), (-|^^2)^ (-1^2)^ (-i^''), and (-x^). 6. (+ f ^), (- I ^), (+ I ^\ (- 2 ^), (+ I x), and (+ ^). 7. (+ 3 a3), (- 7 a3), (- 8 a«), (+ 2 a^)^ and (-11 a^). 8. (+ 4a2?>2)^ (_ aH2), (- 7 a^h^), and (+ .5a2&2). 9. (+7ahcd), (+ 2ahcd), (-l.labcd), and (-4.1aZ>ctQ. 10. + (ft -^ 6'), - .01 {b + c), + .7{b + c), -10 (b + c), and + i(b + c), 11. +10(,.— 2/)3, -(x'-2/)3, +m(x-yf, -2{x-y)\ and — 3 (ic — ?/)^. 17. If the monomials are not all like, combine the like terms, and write the others, each preceded by its proper sign (Art. 14). Example 1. Find the sum of (+ 7 k), (^- 3 h if), (- 2 x), (-5 6 y% (+ 4 X), (-8 6 /), (+ 9 x), (+ 6 1/), (+ 1 1 x), and (- h y^). ALGEBRAIC ADDITION. 23 Ilzplanation. For convenience, write the expressions so that like terras shall stand in the same column, as in the mai-gin. The sum of the terms containing x is -f 29 X, and the sum of those containing b y^ is — 10 6 y"^. Hence, the result is -H 29x- 10 6 2/'*. Process. 4- 7x4- 3 6 1/2 - 2 X - bhy^ + 4 X - 8 6 1/2 + 9x-f 6 1/2 + llx- 6y2 + 29 X - 10 6 1/2 Example 2. Find the sum of -f .05 (a -|- 6), - .01 (m + n) 4- 7 (a + 6), - 3 (m + n), - 1 1 (a + 6), and -f 10 (m -h n). Explanation, (a -f &) and (m + w), enclosed in parentheses, are treated as simple symbols. The sum of the like terms containing (a+/>) is — 3.95(a-|-ft). The sum of tiie like terms containing (m + «) is + 6.99 (m + n). Process. -f .05(0 + 6)- .01(m + «) + 7 (a + 6)- 3(m4-w) - 11 ((! + ?>)+ 10(m + yi) — 3.95 (a + 6) -f 6.99 (m + n) Exercise 8. Find the sum of : 1. (+ .3 X), (+ .5 y\ (+ .01 .>), (+ 3 y\ and (- 7 x). 2. (4- I a), (- J a J), (+ § a), (-h ^ rr ft), and (- | ..)• 3. (+ 5 (-2 ./2), (- 2 a«./;), (- 2 c2.r2), (+10 a^ x), (+ 8 c2 x^), (-4a«x), (-4c2.x'2), and (-f 4a8a?). 4. (+ I .^), (- J a 5), (+ V •^'). (+ 1^5 ^^ ^'). (+ A^^)> (+^^'). (-HJ«^). (-i^^), {-i^^''). and (+§«6). 5. 7a, - :^x-yl 8a, .3 (x-?/), .03Cr - y), and -.la. 18. A Polynomial <>i Compound Expression consists of two or more terms. 24 ELEMENTS OF ALGEBRA. Example. Find the sum oi 8ax— .ly + 5, .7ax + y — am — 9f and - .3ax- 1.02 y + 5p- .3. Process. Sax — .1 y +5 .7 ax + y — am — 9 -.Zax- l.02y ~ .3-t-5p 8Aax — .12 y — am — 4.3 + b p Hence, in general, To Add Polynomials. Write the expressions so tliat like terms shall stand in the same column. Find the smn of the terms in each column, and connect the results with their proper signs. A polynomial may be regarded as the sum of its monomial terms. Thus, the sum of the terms (-f <?), (— h), and (—3 c) is « — ft — 3 c. Hence, the sum of two or more polynomials whose terms are all unlike is expressed b}'^ writing their terms ivith their respective signs. Thus, the sum of a — b, c — d, and m. + n — x is a — b + c — d + m + n — x. Exercise 9. Find the sum of : • 1. 2x-{-y, bx+3y, — 3x — 2y, and — 4 a^ + Sy. 2. 5x + Sy+3a, —7x + 4:y — 8 a, and 2 x — 3 y. 3. 3 6-3^, 2 c - 2 ^, 3c -71), and 4:h-2c + 3x. 4. 14a + x, I3h-y, -11a + 2 y, and ic - 2 a — 12 &. 5. ax — 4:mn + hd, hd — a x — 3 mn, 7 m n — 3 ax + 3hd, and 5mn — 30 ax — 9hd. 6. a — h, 21) — c, 2 c — d, 2d — 3(f-\-n, and m — n + x. 7. a b c -\- 3 a b m — 5 c m, 3cm+ 11 abm-\-9abc, 90abm — 21cm — 31 abc, and 3cm — 51 abm-\- 13 abc. 8. T/i + n + p, m — n—p, m — n+ p, and m -\-7i—p. ALGEBRAIC ADDITION. 25 and — a + 6 + f 4- rf. 10. 1.25 a 6 + 1.1 c + 99 h, ami 3 ah ■{■ 2.2 c -\- 1.01 i. 11 ia^iah + .QV', la - ^^^ « ^ " 1^ ^'. I ^' + i^^2> f .6 ^, and .1 a - 1.01 « h 12. J ?/|2 - .2 wi -f J, .1 wi2 + .01 m - 2, m^ _|. 3 j ^^^ ^^ _ ^^ and J m — 5S m 71 — 1 J w^ — 2|. 13. xy — ac, Sxi/ — 9a(\ —7xy + 5ac+lMcd, 4 xy -^ 6 fl c — .09 f rf, and — x y — 2 a c + c d. 14. .5 r(3 - 2 r|2 h - I b\ lu^b- .75 a l^ + 2 b^ and 15. 3 (m - nf + .3 (a: + ?/)^ .4 (m - 7/)3 - .2 (x + y)\ .7(m-n)8-3.03(a; + i/)^ and 5.1 (m- 7^)3 - M 1 (.aH-.#. ' 16. oa^V^-^a^b^-\-x^y-^xf, 4a^b^-7 a^ b^-'Axf + (jx^y, 3a^l^-^Sa^b^-3a^y+5x y\ and 2 a^lfi ^a^JJi ^ 3 a^ y — 'S X if. 17. |aa.-2-§/^2 4. |.,3y^. j8^ 3^.x^+ |.r^?/+ 7.5^2 + J fcs 2 ^ .^2 ^ 3 ,3 ,j _ 1 ,,2 _ 1 2,8^ j^.^a Jj. a x^-V\x^ y 18. ac^ ^ \a}?' -V In"^ - \a^b - \abc\ \a^ c, \nn j^lb^^laV^^ \b<?-\-2ahr + \\\?c, and 1.1 ^rc- 1 2ac2 + i^^c- |/>6-2l l..Sc3+ 1.23 r/ 6c. 19. 3.1«8-4.2a?2+1.2a; + 1.7, 2.22 a,-3- 1.2 a^ + 3.33 x- - 10.09, 2 a^ + 7 a^ - 2 .r + 1, 3 ./;8 + 1.22 a^» + 12.12 - 1.33 a;, and 11.1 1 j? + 5.55 x^ - 0.2 u.- + 3.77. 20. «H2c3 + «2^^,2 ^ 3 ^^2/^3 ,5 13 ,,3^2,;3 _ 1 4 ,,2^,3^8 + 1.5 a2 63 c2, 1.5 «2 63 c2 - 1.9 ^3 }?• c^ ^ 1 3 ,,2 ^3 ,5 and 1.7 a3 62(^3 _ 1.2^(2^^5 4. 101 a^ftSca. 26 ELEMENTS OF ALGEBRA. 21. 2c'"+.la" + 3&c, .5c'" + 3.9ft"+2.02&c, c'" + 2.09a'» -|&c, and i& + f Z^c- 3.03. 22. a & — a:' + |- a??/, .1 X -{- .01 a ic — 2.02 a 6, ^ ax ■\- ^xy — ^ah, 6ic ~ 1.01 ax + .la;, and ■J'^' 2/ + j a x -\xy, 23. 3 ^ + .ly- 7.01 ^ + 6.01 ?/ + .2 a + 3 ^ - 1.5 c; -.8^ + 9.01a + 3.03?/ + ^-4.04?/- 2.01 2;+ 2.2 a -y. 24. 3 a 6 4- 9 - ic2 ?y, a;3 3/ + 3 a; 2/ + 5, 6 a;^/^ + 4 a;2y — 3 icy, 10 o^y + 1 + 3 a; 2/2, and 17 — 3 x^y — 2 a;^y. 25. .5 {m - 3xy, - | (m - Sx)\ .75 (m - 3 x^, and — 1.25 (>yi - 3 xy. 26. ^2 + Z;4 4. c3, _ 4 tt2 - 5 c^ 8 a2 _ 7 &4 + 10 c3, and 6 &4 - 6 c3. 27. 3a2-4a& + &2+ 2a + 3&-7, 2^2-4^2 + 3a -56+8, 10 ab+ Sb^ + 9 b, and 5 ^2 _ 6 a & + 3 ^2 + 7 a - 7 & + 11. 28. x^ — 4:a^y + 6x^y'^ - 4:Xy^ + /, 4:S(^y — 12x^y^ + 12 xy^ — Ay\ 6 x^y^—12xy^ + 6 y^, and 4:xf — 4:y\ 29. a3+ «62+ ac^-a^b-abc-a^c, a?b ^- b^ + bc^ — ab"^ — b'^c — ab c, and a^ c + b"^ c + c^ — a b c — b c^ — a c2. 30. 5 a^ - 2 a^b + 9 ab^ + 17 b^, -2 a^ + 5 a^ b -4:ab^-12b^ b^-4:ab^-5 a'^b-a^ and 2 ^2 6 -2 a^-Ob^-ab"^. 31. a;»« — ?/"+3< 2ic'"-3/-a, and a;'» + 42/"— a''. ALGEBRAIC SUBTRACTION. 27 CHAPTER III. ALGEBRAIC SUBTRACTION. 19. In Art. 12 it was shown that to subtract a positive imuiber means to count so many units in the negative di- rection, and to subtract a negative number means to count so many units in the positive direction. Hence, the addi- tion of a positive number produces the same result as the subtraction of a negative number having the same absolute value. Thus, 4-3+ (+6) = + 3 + 6 = 9. +3- (-6) = + 3 + 6 = 9. Also, the subtraction of a positive number produces the same result as the addition of a negative number having the same absolute value. Thus, + 4-(+6) = f4-6 = -2. +4+(-6) = +4-6 = -2. We observe that the subtraction of one number from iinother produces the same result as counting or measuring from the place occupied by the subtrahend to the place occupied by the minuend. Thus, Subtract — h from + a ; also + a from — 6. -a^ -<r -b a-b +b a<r IB D 28 ELEMENTS OF ALGEBRA. Explanation. Suppose the algebraic numbers to be accurately measured on the line of numbers C D. We start at B in the positive series, count b units in the positive direction, and arrive at D ; and the distance from A (0) to Z> is equal in absolute value to A D in the positive direction. But in counting from ^ to Z> the absolute value is the same as the absolute value in counting from C (the subtrahend) to B (the minuend), and we have counted in the direction opposite to that indicated by the sign of the subtrahend. Thus, C B = -h {+b)-\- (+a) = a + b. That is, (+ a) -(-b) = a + b. Subtracting + a from — b gives the same result as counting from a in the positive series to b in the negative series, and the distance from 5 to C is equal in absolute value to B C in the negative direction. Thus, BC = + (-a) + (~b) = -a-b. That is, (— b) — (-{- a) = —b — a, OT — a — b. Hence, Algebraic Subtraction is the operation of finding the dif- ference from the subtrahend to the minuend. To subtract — 5 a from + 2 a is the operation of finding hoio far and in ivhat direction we must go to pass from 5 a in the negative series to 2 a in the positive series, and is found, by counting from — 5 a to + 2 a, to be 7 a units in the positive direction. That is, + 2a - (- 5a) = + 7a. To subtract + 5 a from - 2 a, we count from 5 a in the positive series to 2 a in the negative series and pass over 7 a units in the negative direction. That is, - 2 a - (4- 5 a) = - 7 a. These differences may be found by changing the signs of the sub-' trahend and proceeding as in addition, as shown by a comparison of results. Thus, Minuend. Subtrahend. By Addition. + 2a-(-5a) = + 7a >| j^+2rt + (-t-5«)=4-7a. -2a- (+5a)=-7a ' I - 2 a -f (- 5 a) = - 7 a. + a - (- &) = + a + 6 f ''''' ) + a + (-{- b) = a + b. - b-(+ a) =-a-b ) \^- 6+(-a)=-a-6. ALGEBRAIC SUBTRACTION. 29 Hence, in general, To Subtract one Algebraic Number from another. Change the sign of the subtrahend, and add the result to the minuend. Notes: 1. Algebraic subtraction considered as an operation is not distinct from addition ; for it is equivalent to the algebraic addition of a number with the opposite algebraic sign. It includes not only distance but direction, and direction depends upon the sign of the subtrahend and which number is consid- ered the minuend. 2. Algebraic subtraction is not in all cases diminution. Thus, 8 ~ (- 2) - 10 ; also 2 - (- 8) - 10. E.xAMPLE 1 . Subtract + 3 o* 6 c w* from -f 10 a* 6 c 7/1^. Solution. Changing the sign of the subtrahend, and proceeding as in adtlition, we have + 10 a* 6 c m^ -f (— 3 a* 6 c m*) = + 7 a^bcm^. Example 2. Subtract + 27 (x^ - i/«)» from 13 (x^ - i/)». Solution. Treating (t* — y^y as a simple symbol, changing the sign of the subtrahend and proceeding as in addition, we have 13 (x2 - 7/)« + [- 27 (x^ - !/«)»] = - 14 (z" - y«)». Exercise 10. From : 1. +OaHc take —aHc, —Ual^xy^ take +19 al^xi/. 2. +:tVtake-./.y; +99 mnp^rst^^ take ■}-99mnphst^^. 3. —10 axy take —axy\ x'^ take —he. From the sura of: 4. — 11 a.-^, + .5 d^, and + 1.25 J" take -f 5.5 0^. 5. ah^, — Sabc^, and + .3 ah(^ take the sum of -abc?^, + SMabc^ and - IMabc^. 6. lOS mnp^^, —lO.Smnp^^, and +vinp^^ take the sum of -- 10 771 np^^, + 33 mnp^^, and — 108.1 m np^^ 30 ELEMENTS OF ALGEBRA. 7. 5 {x + y), — 2{x-\- y), and + {x-\-y)y take the sum of - {x + v/), + 6 {x + 2/), and - 2.5 {x + ?/). Find the aggregate value of: 8. + 17 a x^ -(-5a a:-3) + (- 24 a ^'S) - (+ a j:^). 9. + 19 a ^ ?/2 + (+ a ;2; 7/2) — (— 5 a x ?/). 10. + x-2y + (- x^y) - (+ x^y) - (- ^2y) + (_ 1 .^^2^/) - (+ 3 x^y) - (- 10a;2y) + {-bx^y). ' 11. + n« + ^)' - [- -1 (« + ^)'] + [-(« + ^)'] - [+ I («■ + ^)'] + 10 (a + ?^)2. 12. I .- + (+ 1 X-) + (- .1 X-) - (+ ,2^- ^") + (- 1 x^ + (+ 1. ix'')-{-2>ix'y 20. Example L Subtract 2 ah + f) a^ if - U a^ - 1 y^ from 15 a3 - 8 1/ + 23 a3 t/S. Process. Minuend ] 5 a^ _ 8 ?/3 + 23 a^ i/S Subtrahend, with signs changed + 14 a* + 7 ?/^ — 5 a^ ?/^ — 3 a 6 Difference 29 a^ - ^/^ + 18 a^ ^/S - 3 a 6 Example 2. Subtract 3 a; ?/ + n - 5 a^ 6 + 5 jo'' from 5 ic i/^ - 3 a^ 5 + 3 771. Process. Minuend 5 xy^ - 3 a'^h -\- Sm Subtrahend, with signs changed -3a: 7/^+5 a- 6 — n — b p^ Difference 2xy^-\-2a^h+'3m — n — 5p^ Hence, iu general, To Subtract one Polynomial from another. Change the algebraic sign of every term in the subtrahend, and add the result to the minuend. Note. It is not necessary that the signs of the subtrahend be actually changed, we may conceive them to be changed. ALGEBRAIC SUBTRACTION. 31 Exercise 11. Subtract : 1. 5x~3y-f22 from 3 x -\- y — z. 2. X — y •{■ z from — x — 2 ij — 3 2. 3. a — 6 4- 20 6' from — a — h •{• 10 c. 4. J^ — iy — i^ from \x -\- y + z. 5. lx-\y-\-^zhom-lx+t^y-^^z, 6. J^-iy-i from -Jx--Jy+ J. 7. a3 _ 4 a2 J ^ 2 i c + 5 from 3 a^ - «U - 5 c - 5. 8. a^y — Z ah X -\- 2 X y^ — \ from S x^y — abx-^ 2 xy^. 9. ahcxy -\-2ahy — 4ihx from 2abcxy — ab y -\- bx — 'S. 10. acy — bxy + aJc — 1 from bxy — 4:acy — abc -fa. 11. .4a:*-.3a:3^.2ic2-7.1a; + 9.9 from a;^ - 2.10 a:^ + .2ar»-:.07a.' + .9 12. 1.2 /-s - 1 4- X + 1.1 a;4 + 1.7 a^ + a from \-x -.l.r* + .2.'/^-.3A'8 4-a. 13. \ vin^ ^In^- yi^ + I w2 from | /7i3 - J r/i 7i2 - ;,2 14. .125 m3 - M^ ,n vi2 _ .33^ ^ ^2 _ (5o| ^3 _^ ^. |Vom g Wl^ — I Wl n^ — I ?//,2 7J, -I- J 7^3 15. J ,;i2 _ I y _ I ^ 4. J ^ |.^Qj^ ^ W2- ^ y 4- f ?l - i .^. 16. a^hc + xf -Ic from 3^ a2ftc + 2Jaj?/"- 4f c. 17. lOc — a-h -^ b(l-\- 6rt-15c4- 3(Z from 25«-5 — 5c4-8r? — 20rt. 18. af" — 3 a:" if — if from 4 x*" + .c* y" _ af . 32 ELEMENTS OF ALGEBRA. 19. — .9 a"* x^ - .3 a 63 a? + .6 + .03 IT c x^ from .9 a'^x^ + 1.3 -2ah^x-\- AlTcx^ 20. From the sum of lo?-\h^ +\ c\ J cv^ - 3^ c\ and 2| 63 - I a2 _ i| c^ take ^i a? - ^\ 63-4 c\ 21. From the sum of 3 ic3 _ ^2 ^ _ ^^^ 7 ^3 _ il ^2;3 -f- llj y^ oc, and 11 ;;c2;3 _ gi ^3 _ 2 1/2 a; take a;2;3 _ 25 ^/^j; 4- 1 ^3. 22. From of + y"" take the sum of 11 ^-^ + y^ — z, — 6 ^" — 5 /* — 3 ;2, and - 5 ^" + 3 2/"* + 4 z. 23. Add the sum of S^y- .3 ^/^ and 5-3y + 2.7 2/3 to the difference obtained by subtracting 3 + l^y^ — .5y from 1—2/3. Queries. Why change the signs of the subtrahend in subtracting ? Wliy add the subtrahend, with signs changed, to the minuend ? Does the use of the signs + and — in Algebra differ from their use in Arithmetic ? How ? Miscellaneous Exercise 12, 1. From m^ — n — 1 take the sum of 2 n — 3 + 2 ?^3 and 3 ??i3 — 4: + 5 'n? — n. 2. From the sum of 1 - 8.8 y + .9 a^ and 1.1 x^ + S x^ — .2 2/ — 1 subtract 2 x^ — x^ + 5 y. 3. Take x^ + x — 1 from 2 x^, and add the result to -2a^-x^-x + l. 4. Take a^ — h^ from ah — h^, and add the remainder to the sum of a6 - ^2 - 3 6^ and ^2 + 2 62 5. To the sum of m + 7i — 3 /? + 5 and 2 m + 3 ?i + 5^ — 3 add the sum of m — in — 7 p and 5 ^ — 6 ??i — 2. ALGKBRAIC SUBTRACTION. 33 6. Take 3 a-^"* - 2 x-2" f^ - y^-^ from 3 f/—' + 2 x^'^y^ 7. Take 2x^y^-*6z^-\-2y^-V z-1 from a;iy*+2;--4yi 8. Take .8 h^ y^ - A a? x^ + .3 rf from .4 a? x^ + .3 c - 1.2?)tyi 9. Take f xt-f a;*y*H- 33j?/§ from f j:;^ + |-^*?/* + Jyi 10. From the sum of .7 c y^ — .4 a x -f .5 h, .04 6 — | cy» + i w,-|a.r+ Jcy§-|, and -yi a x- .23 6- .8 m + .3 take the sum of .55 a ic + J /?i + ^^ and .33 in — 1.1 cy^ + .67 ?«. 11. Find the sum of «"• - 7 6* -h cp and | 6" + | a"*, and subtract the result from cp — 4:n. 12. From a* - 2 c" - af* take the sum of J «"• - ^ i" - X' and i a"* + J 6" — y" — af. 13. From - a*-6*-c'-^« take the sum of Jft'"+ §6* - I C, ^5 a"* - \l c^, and V ^^ + ^'■ 14. From 3 (a^ - ^y - (o^-a + ff take § (^ + 2^)* - c^ + 3J («^ - ^)*. 15. From unity take 3 a^ — 3 a + 1, and add 5 a^ — 3 a to the result. 16. Add 3 ^- - 7 a:" + 1 and 3 a;8- -f ^ - 3, and diminish the result by x^"* — 2. 17. From zero subtract I a^ — ^ x -\- 2. 18. From .3 m' - 1 + J w take 5 n^ - 2.7 m^ - J n, tlien take the difference from zero, and add this last result to - 5 n^ 4- 3.3^ m^ -f n. 34 ELEMENTS OF ALGEBRA. 19. What expression must be subtracted from 10 y'^ -]- y - 1 to leave 3 y^ - 17 y + S ^. 20. What expression must be subtracted from a — o x + y to leave 2 a — o x -\- yl 21. From what expression must a^—bah — lhc be subtracted to give a remainder b a? -\- 3 ah -\- 1 h c1 22 From what expression must a^ h^ — b^ c^ + 6 a"" c** be subtracted to leave a remainder b^ c^ — 6 aJ^ c" ? 2*3. To what expression must | ft^ + 2\a — 1^ a^ — 3 be added so as to make 2\ a^ — 2\ a -\- 3\ a^ -\- ^ ^ 24. To what expression must b x y — lib c — 1 mn be added to produce zero ? 25. What expression must be added to 3 ^" — 3 ^""^ + 2 to produce ^" + x^"~^ — 6 ? 26. What expression must be added to m a?"* — ^" + 2 to produce m a?'" — 2 ? 27. From the sum of .6 {x + y)^ + .3 a" + ^'", | a** x"^ — c^ ~ I (« + ?/)2, and I (^ + 2/)^ ~ I ^" ^''"j take the sum of .3 (x + i/)^ - I a" ;:c^, I «" ic'" - 6.5 (ic + y)^ + c^ and ro (^ + .?/)^ + 3.3 a" ^"^ — 3. Algebraic Subtraction may be defined as the operation of finding a number which added to a given number, will produce a given sum. The sum is now called the min- uendy the given number is the subtrahend^ and the required number is the difference. ALGEBRAIC MULTIPLICATION. 35 CHAPTEK IV. ALGEBRAIC MULTIPLICATION. 21. Evidently dm x 6n = 5 x (^ x m x n = SOmn. Hence, in Algebra, the product is tlie same in whatever order the factors are written. a X a X a X a or aaaa is written a*, and shows that a is taken four times as a factor. aXaXaXaxaov aaaaa is written a^, and shows that a is taken five times as a factor, a X a X a X ton factoi*s, or a aa to ?i factors is written a", and shows that a is taken n times as a factor. Hence, An Integral Exponent shows how many times a number or term is taken as a factor. a^ is read a second power, or a exponent two, or a square. a' is read a third power, or a exponent three, or a cube. Hence, A Power is the product of two or more equal factors. The degree of the power is indicated by an exponent. a? = a a a, and a^ = a aaaa a . Hence, a^ X a^ = a a a a a a X aaa = a» a" = a a a a .... to n factors, and a*" = a a a a .... to m factors. Multiplying the second expression by the first, we have, rt"* X a* = aaa to m factoi-s X a a a .... to n factors = a a a .... to (m + ?i) factors = a*"*"". In which m and n are a?iy numbers 36 ELEMENTS OF ALGEBRA. whatever. Similarly for the product of more than two powers of a factor. Hence, The i^owers of a number are multiplied hy adding the exponents. If the multiplicand and multiplier consist of powers of different factors, we use a similar process. Thus, 3m^ X 2 111^11? X 5 m'^n^ = o X 2 X 5m 771 m m m mmmnim X n n n n n a"fe"* X a^lf =aa a .... to n factors x aa a .... to p factors Xbbb to m factors Xbbb .... to r factors = aaa .... to {n-\-p) factors x bbb .... to (m + r) factors =: a""*"^ 5'" '^'■. Hence, in general. To Find the Product of Two or more Monomials. To the product of the numerical coefficients annex the factors, each taken with an exponent equal to the sum of the exponents of that factor. Notes: 1. When no exponent is written, the exponent is 1. Thus, a is the same as ai, & as fti. The exponent is used to save repetition. 2. We read a^, a square, and a*, a cube, because if a represents the number of units of length in the side of a sqiiare, and the edge of a cube, then ffl2 and a3 will represent the number of units in the surface and volume of the square and cube, respectively. Illustrations. 11 mi9 X 10 m}^ =11 X 10 mi^ + lo = no ni^^. 3a'^bcmX2ah'^cmX 5abc^m^ = '3 X 2 X 5a2+i+i6i+2+ici+i+«mi+i+2 = 30 «4 ^i c* m\ 3a^fe*c3 X 4rt X 6'cJ = 3 X 4 a' + ift^ + J c3 + ^ = I2a^bc*^\ S'x^i/" X 2^x-3 X x^y"" = 25 + ^x2-3+57/" + " = 2 x*i/2". ALGEBRAIC MULTIPLICATION. 37 Exercise 13. Find the product of : 1. 0? and 7 a?^ ; 3 a a; and ^ c^t^ \ a? ha? and 2 a^ W a?. 2. ^xyz^^ and Is^t/^mn; ^abcdm^7i^ and ^a^lf^c^d^mn. 3. |^/2^.3yaud |a6366y" + '; 3a3a:«3/7andf ai«^a:8 3^*2. 4. 3 a a:^y^ and 10 a^^xy^^ ; J x"* if and | rr* y^. 6. 3aftca;^V and f rt-^j^c^^^ \a'hh^xy and | a Z>iOc V^^"*. 6. f a'-fc-af / and .2 a^ h^ 2^ f , a? f and a^^i/S. 7. a'-iz-and tt-6'"; a-^-^V^^and 5.7a:-i3/-i. 8. ahx]/^ and a?-}?x^y\ a"'+'i>'* + '* and a'"-''/;"-'". 9. .55a;-' + '//-'' + ' and .5 a;''+«/-''; .3 a2-"';j;8- n ^j^^ a* 2:"; 5a-H".r^ and ^oa\hx'^\ 10. 2«^2:, a-?/, «2//, a^j^i/, and a ^. 11. a*-, fc", 3c', a', 5', c*", and c?'. 12. 2c'"</, «c", r/22:", a"* a;*", and ci 13. ai VI x^, a^ ni x^, a m x y, and 2 a^ 71^ x^ if. 14. a^z^, «^?/^, ala;"i, a^y~^, 5fa"la;^, and 5^2:^yV 15. fa^ma:", jTnta:*, .Sarr"*. 5.1 w', and a-^'x-', 16. 3y*, a-^y'^z^, (^ b"", a^b', f «^.7/", and ^a^J"'^-. 17. (a 4- ft), 5 (« 4- 6)2, 3 (« + 6)^ ^ (^ + &)^ and (a + bf. 18. (a + 6) (c + (if, {a + bf, 3 (c + df, and (a + 6)7 (c + rf)2. 19. 3 (a + 6)- (a: - y)-, J (a + 6)«, and | (a: - yf. 38 ELEMENTS OF ALGEBRA. 22. Algebraic Multiplication is the operation of adding as many numbers, each equal to the multiplicand, as there are units in a positive multiplier ; it is also the operation of subtracting as many numbers, each equal to the multi- plicand, as there are units in a negative multiplier. Hence, The multiplier shows that the multiplicand is taken so many times to he oAded, or so many times to he suhtracted. Thus, (+ 6) X + 4 = (+ 6) + (+ 6) + (+ 6) + (-f 6) = + (+ 24) = -f 24 (_ 6) X + 4 = (- 6) + (- 6) + (- 6) + (- 6) = + (- 24) = - 24 (+ 6) X - 4 = - (+ 6) - (4 6) - (+ 6) - (+ 6) = - (+ 24) = - 24 (-6) X-4= -(-6)-(-6)-(-6)-(-6) = -(-24)=:-f 24. ■ The sign of the multiphcand (6) shows that the product (24) ii^ in the positive and negative series of numbers, respectively; and the sign of the multipl ier (4) shows that the first two products are to be added and the last two are to be subtracted. Hence, The sign of the multi-plicand shows what series of numbers the product is in, and the sign of the multiplier shows what is to he done ivith the product. Law of Signs. The product of two factors is positive when the factors have like signs, and negative when they have un- like signs. Since -2x-2 = +4; -2x-2x-3=:-h4X-^3==-12; -2x-2x-3x-4--12x-4 = + 48; _2x-2x-3x-4x-5 = + 48x-5 = -240; etc. Hence, The product of an even number of negative factors is posi- tive; of an odd number, negative. ALGEBRAIC MULTIPLICATION. The change of signs may be illustrated as follows 39 > + ax-Sr^ K. ox+y »^y»w.s ax-3 V QX-^3 VCATT ClX-2 < ax+i Q X-/ ^ ax+/ ^ A * k -QX*Jl -ax+/ . , -ax- A -fly-3 -CtX-i' ^ 7 ... - -ffX+J^: •CTK^S^ k ■ -ax-s -| -< z^ Let the measuring unit be represented by a. From A (o), the starting-point on the scale, measure toward the right and left. The products of + a and — a by the factors from -f- 5 to — 5 are : aX-l-5, aX+4, ax+3, rtX4-2, ax + 1; a X - 1, a X - 2, a X - 3, rt X - 4, a X - 5 ; -aX+5, -rtX4-4, -ax+3, -ax+2, -aX + 1; -aX-1, -aX-2, -aX-3, -ax- 4, -aX-5; respectively. The directions taken by the products are shown in the figure. ninstrationB. x«y» X -x*z X -^j/z^ X -^xz^ X - 4 yz^ = -f- f X ^ X 4x«j/*2»o X^y-* X - fx-y-Z X - yz-'- X - X~^'* = ~ ajm + n-Sny-n + n+ljl-r Exercise 14. Find the product of : 1. 5 a, — 3 /), 7 c, — 2 a*, — 11 a^, and a; a^x, —ay^, a a^, and — xy. 2. ahx, —ay^, —a X, and a^a?\ —al^, —hc^, —cd?, — a, — a^, — a^, and — 5 a* 40 ELEMENTS OF ALGEBRA. 3. — a, he, — 1, ^, 1^ a^, ^x y, and 75 a; ax, ex, — m X, — 2/**, and .3 ?/i. 4. ^abc, —d, ax, —1, and ^axyz; a'^af, af^y*", a'^V, and a b. 5. —a^x, 3x, ah^, ay, az, and axyvw; axy, —^a^V, and - SJa^&'a:"*?/". 6. - a'^hc, 2 h'^cd^ - .5 a^ccl^, - f^ a-^H-^^c^^d'^^, and a h^ d*. 7. -1, a- 3, a^7^ a 0^-5, aio^-3^ a-H-^a^, and -Mr/2 8. aaP, — a\ — 1, .3 a x, and — a^^/^; ^^^.^ _ ^|^ ^|, and — a^ a^i. 9. — my, mx, — mn, — xy, and it* 3/3; 3 aa Jt and — .7 «i Z>i 10. a", a^", a^", a^**, and a^". Express the result in two ways. 11. 2^ij-'x-^, mx'^y'^', -3" ^"2:-!, and -2-^po(^f. 12. 32", -23« X 3'^^^ 32«, - 23" X 3*«, S^" x 2«, and — 26" X 3« + i. 23. Example. Multiply a + & by m ; also a — fe by m. The symbol (a + h) m means that m is to be taken (a + V) times. Hence, Process. (a + h)m — m -\- m ■\- m + taken a + h times = {m+m+m+ — taken a times) 4-(7w + m + m+ — takenft times) ~ am -\-l)m. (1) ALGEBRAIC MULTIPLICATION. 4l Also, (a — h)m = m-\-mi-in-h .... taken a—b times = (m + 7» + m-f- taken a times) — (m + m+m+.... taken K times) =:(rnX a)-(mX b) — am — bm. (2) Similarly, (a + ft — c) m = a m -f- 6 m — c m. These results are obtained by multiplying each term of the multi- plicand separately by the mnltiplier. Hence, in general, To Multiply a Polynomial by a Monomial. M%dtiply each term of (lie muUiplicaiul by Ute multiplier ^ and add the resiUts. Exercise 15. Multiply : 1. hc-^-ac-ab hy abc; S aH^ - ^ hh^ - ^ c^ hy -f^a^b^^. 2. 5a^-b'-2c^ by a^b^c^^; .6 3^- .5 2^y^- .32^y^ - .2a^ by .2x^f. 3. j W.2 — ^mii -{- ^n^ by ^mn; x — y — ^x^y^ by xy. 4. fa-^j&2_^^^a62 by|«Z^2. a' - a^lr^-ab hy ah^. 5. 6a2a3- .5a^b^x^^ -\- .2h^2^ by ^ ab 3^; pxT-qx* — r by p2^ r. 6. 3a'"-»-2 6"-H4a'"6" by a&2; .4a— " 5'''-Ja-*'6' + ft3^ by I a-^-^'ft'. 7. a*--3a'"i—4-&'* by a'-J'^"; 2^2:^ - 2iyi + 2^a;iyi by 2^x^yi 8. a?-a2jf-an + 5iby aU^; a;*-2rty^4-a:^i^t-.6y* by xhj~^. 42 ELEMENTS OF ALGEBRA. Find the product of : 9. x^y^-4: x^f+ 4/, xh/, and -2i/; m''^ - 2m^^7V'' + n^\ m~^, n-\ and in" n\ 10. ^^26-4 + 1^6-32: + 1 62 2:2^ laV^, %b-^x, and ^aH^a^. 11. a;3 — 7/F a;3, 2/3^ and —x'^i/^; a—M, a?, 63, a^b^, — a2 jf and — a b^. 12. x^—i/, x^, x^y^, —x^yi, | ?/f , J 2:t, — |- ?y^, and .21 .2_1_ rr^ 7/4. 13. J - .2 6f :2:2 + ;3 7; -^i _ ^1^ J ^1^ _ il ^2^ and J b^ xi 14. 1^--^'" -y «--6-"' + |6, 'Sa-"\ -rjb-'^, and 15. «"+« + f/"6'« + (fc"»6'* + 6'"+", a.'", ft"*, rr", 6"", and 24. Example 1. Multiply m-{-ti by a;+y; also m^n hyx — y. (m + w) (v+y) means that a: + ?/ is to be taken m + n times. Ifence, Process (m + n) X 0- -T-y) = (^' + y) + (^' + 2/) + C-^' + y) 4- • • . • taken w + n times = [(*' + ?/) + (•^■ + ?/) + (x + y) + ... taken m times] + [ (*■ + ?/) + C^' + y) + (^ + y) + taken n times] = (x -{- y) m + (.r + ?/) n = (1) Art. 23, ??* a- 4 m y + n x -\- n y. (1) Also, (m + n) (x - y) = (x-y) -{■ (r~y) + (x-y)-\- .... taken w + n times -■ [(-^ — ?/) + (t ~ ?/) 4- (?' — ?/) 4- — taken m times] 4- [(^ " y) 4- (j" — ?/) 4- (r — 1/) -f — taken n times] = (x ~ y) m-{- (x - y) n — (2) Art. 23, nix — my -\- 71 x — n y. (2) Similarly, (7n -\- n + p) (x-\- y — z) = m x + 7n y — mz + nx + n y — 71 z -\- p X -j- p y — J) z ALGEBRAIC MULTIPLICATION. 43 These results are obtained by multiplying each term ot the multi- plicand separately by each term of the multiplier, and connecting the products with their proper signs. Example 2. Multiply j*-i*+2x^-x-5 by x*-h3x*-\- 5. Process. x«- r«-j-2x«-a:-5 X* + 3 x« + 5 x" - x« -H 2 x« - x« - 5 X* + 3z» +6x*-3ir*~3x8- i5x« + 5 x« - T) j;5 + 10 x*-* - 5 x - 25 xW -H 2 x» -I- 7 x« - « X* - 3 x8 - 16 x» + 10 x2 - 6 X - 25 Explanation. Multiplying each term of the multiplicand by each term of the multiplier and connecting these results with their proper signs, we have x**' — r* -f- 2 x* — x* — 5 x* + 3 x* — 3 x* 4- 6 x* - 3 X* - 15 x« + 5 x« - 5 x« + 10 x2 - 5 X - 25. Umling like terms, for a simplified product, we have x^<> -f- 2 x* — 3 x^ -f 7 x* — 8 x* — 15 x* + 10 x« - 5 X - 25. The process used in practice is shown above. The first line under the multiplier contains the product of the multiplicand and x*. The second contains the product of the multiplicand and 3 x*. Etc. To facilitate adding, write the several products so that like terms shall stand in the same column. Hote. It i^ convenient to arrange the terms of the multiplicand and multi- plier according to powers of some common letter, ascending or descending. Example 3 Multiply f a x + f x^ + ^ a* by f fl^ + § x^ - f a x. Solution. Arrange the expressions according to the descending powers of r. Taking the multiplican<l | x* times, we have x* + a x' -f ^ rt^x^. Taknig it — | n r times, and writing the proiluct so that like terms nhnll stand in the same column, we have — ax* — a*x* - 1^a*T, Again, Uiking it | a^ times, and writing the pnnluct as be- fore, we have ^a^ x^ -h ij^a* r + \ a*. Adding the partial products, we have x* -I- ^ a*, or arranging alphal>etically, { a* -f- x* 44 ELEMENTS OF ALGEBRA. Process, f x2 - f a a: + f a^ — ax^ — a'^ x"^ ~ ^ a^ X + ia^x^ + la^x + ia* Example 4. Multiply - Sx^' + ^y^ - .Sx^^^^ y'' + ^ + 3.3j:"'^ + ^ by — .2 j:™«/«-2 + 4 a;'"-' i/«-^ Process. 3.3 a;*"*/" + 2 — .3 x"" + ^ y'' +^ —3 x"' ^'^y'' 4 x™ - 1 ^'» - ^ — .2 a;"» ,y" - ^ 13.2r*"»-i/«+i -1.2 a:-'"**/-'" - 12.00 x^^ + 'j/an-i - .66a;2'«?/2»,_^ 06a;2"» + i j,2«-i^ g^zm+a^^n-g 13.2 j2m-1^2n + l_l 86^:2 '"y^»- 11.94x2"' + ! ^2«-I_^ (5 a,2m + 2^2«-2 Explanation. Arrange according to the ascending powers of x, as shown. The product of the multiplicand by 4x'"-' ^" ~ ^ gives the first partial product, as shown on the first line under the multiplier. The product of the multiplicand by — .2x'"^"-2 gives the second partial product. Taking the sum of the partial products, we have the product required. Hence, in general, To find the Product of two Polynomials. Multiply the multiplicand hy each term of the multijplier, and add the partial products. Exercise 16. Arrange the terms according to the powers of some common letter, and multiply : 1. r/2 J^h'^-ah by « 6 + ^2 + ^2 . a'^-2ax ^ 4.x^ by «2 + 4 .^2 + 2 a a^. 2. x^ -\- y^ — x^ y"^ by x'^-\-y'^\ x + y -{- x — y by x-^-y -x + y. ALGEBRAIC MULTIPLICATION. 45 3. y— 3 + /y2 by 2/-9+2/2; a^tj — azi-ij^ — a^ by y-^ a. ^ J a;2 - I X- - f by J a;2 ^ I ^ __ 1 . 1 6 «2 4. 1 2 a 6 + 9&2 by Aa-.:n, 5. x^— i/-\- X — fj by x^-^ y^-\- x — y \ ^s^ — ax — ^a^ by \ x^ — ^ax -\- ^a^, + \d^ by 2 a;^ + « 2: — ;| ^3. 7. n7^-Dx'^-x^^-2^-x + 2hyx^-2x-2, 8. 3a2-2a3_2a + l + a* by 3ft2+ 2^3+ 2a + l + ^*. 9. 1.5 2:8 + 1.5 2^2 + .5 a:* + .5 2: + 2:^ + 1 by a^ - .5 a; + 1 + a:* - .5 2:8 10. 1 + 9 a + 5 a3 + 3 a* + 7 a2 4- a^ by 4 a2 - 3 a8 4- a* 4- 4 - 4 a. 11. 4 2:2^24. 82:^3+16/ + 2a:3y _^^ by 2: - 2 y. 12. 2^12- a:3^6 4. 2^6^- 2:87/24. y8 by y«-|- 2:8. 242^2 — 3^ y — X i^ -\- x^ -\- y^ by x -\- y. 13. rt2 _^ ^2 _^ ^ _ ^ 5 _ rt c _ ^ c by rt + & + c. 14. ^2 _^ ^,2 _^ ,.2 4. J c + a r - a 6 by a + fe - c. 15. a6 + crf4-ac4-6c^ by ab-hcd^ac — bd. 16. i2 4. y2 _ 3 3^ _ y2 by 2 a: + 2 3^ - 2 (2: - y). 17. 3 (m + n) — .1 X (a + h) by a - b -^ .1 (m ^ n). 18. a2f^-\-bx''-\-r by a2:* + 62;"+r; 2;^ + yi by 2:^ — y^- 19. a- + 6" by a'" + b"; oT 4- 6* by a" - 6*; 2:2 _,_ j by a:i + bl 46 ELEMENTS OF ALGEBRA. 20. Sx"^-^ - 2/-' by 2x- S'f; ax''' + &^"+ ahx by a x^ — bx^ — 1. 21. 'Sa^^'x+'Sa^y + a''' by a'"- a" + 2:; x^-y-i by 2^2 — y. 22. .2ai-.3&t hy.2ai+.3bh^xi + xUji-hy^ hy x^-yk 23. a;^ ?/~t + y~^ + x^y"^ + a;^ by x^ — y~i. Find the product of: 24. 1 + 2;, 1 + 2;^ and 1 + x^ — x — a^. 25. a; — 2 a, X — a, x + a, and a: + 2 a. 26. 3 2; + 2, 2 a: - 3, 5 :z: - 4, and 4 a; - 5. 27. ^2 — :r + 1, a;'-^ + :r 4- 1, and x'^ — x'^ -{- 1. 28. rr^ — a x-{- a?, x^ + a x ■}- a?', and x^ — a'^ a;^ + a*. 29. « + 6, a - 6, 3 a + &, and a^ -2o?h - a}?' \ b^ 30. rr + &^ «"*-&", ft''"+a"*6"+?)'", and a2'«_,^«^H + ^a«^ 25. A Binomial is a compound expression of two terms ; SiS, a — b; ab + 2b\ In each of the following products, observe that : 2a; + 3 2 a: + 3 2x + 5 2a; - 5 4a;2+ 6 a; -~ 4x2+ ^^^ 10 a; +15 -10 a; -15 4x2 +16 a; +15 4^2- 4 a; -15 2a;-3 2 a; -3 2x + 5 2a; — 5 4a;2- 6 a; 4^2- 6 a; 10 a; -15 -lOx+15 4x2+ 4 a; -15 4x2-16x+15 ALGEBRAIC MULTIPLICATION. 47 I. Thejirst term is the common algebmic term of the binomials multiplied by itself, or the square of the common algebraic term. II. The second term is the al«;ebraic sum of the other two terms of the binomial expretssious multiplied by the common algebraic term. III. The last term is the algebraic product of the terms which are not common to the binomial expressions. Hence, To find the Product of two Binomials, having one Common Algebraic Term yidd toy ether the nfpiare of the common tertn, the abjebraic aum of the other two tervis multiplied by the cmiimaii terniy aiul the algebraic proditct of the terms which are iwt common. In general, {x -\- a) (x ±b) — 2^ + {a ±b) x ± ab (1) (x-a){x±b) = x^-\-{-a±b)x^^ab (2) In which a, b, and x represent any numbers. Hotel: 1. It is of the utmost importance that tlie student sliould learn to write tlie products of binomial expressions rapi<lly, by inspection. 2. To square a monomial, multiply the numerical coefficient by itself ^ and multiply the ejptment of eacli letter by two. The proof is evident. Thus, the square of 2 a* 6» =- 2 X2 a* ^ * A* >< « - 4 ofta-. Also, (36-"ar«)a = 3 X 36-»x2a*« ^a - 96-2««a;2m. Examples. Write the product of the following by inspection : (2 a; + 7 y) (2 I - 5 i^); (a - 9 6) (a - 8 6) ; (a - 6) (a + 1 3) . Solution. Squaring the common term, we have 4x^. Taking the algebraic sum of the other two tenns, + 7 y and — 5 y, we have + 2y. Multiplying this sum by 2 a;, we have + 4a:y. Taking the algebraic product of the terms not common, + 7 y and — 5 y, we have — 35 y*. We thus obtain 4x^ -i- 4 xy— 3b y^ for the product. Similarly, (a-96) (aSb) = a«+ (-96-86) X rt + (-96) X (-86) = a3-17a6-|-72 6^. Also, (a-6)(a+13) = a2+(-6 + 13) X a-f (-6) X (4-13) = a2 + 7a-78. 48 ELEMENTS OF ALGEBRA. Exercise 17. Write, by inspection, the products of the following : 1. (a -3) (a + 5); (6+6)(&-5); {x + 4) {x + S) {x - 4) (0^ + 1) ; (^ - 7) (x + 2). 2. (x - 8) (^ - 6) ; (a + 9) (a - 5); {a- 8) (a + 4) (2x-4:) (2 a^ - 5) ; (3 ^' + 7) (3 ^ - 5). 3. (0^3-37/2) (^3_ 4 2^2). {x-7y)(x + Sy);{a"^-l){a-+2) (3a:5-5)(3a:^-4). 4. (2 a2 2/3 + 4) (2 ^2 f _ 8) ; (3 a a; - 4) (3 a 2: + 7) (a:3 + 3 a) (a;3 - 4a;) ; (ai^ - 3 a2) (a;^ + 2 a^). 5. (2a; + a)(2a;-2a); (2:c"+ 5a)(22;"-3«); {Sx-2y) (S X + y) ; {- 6 m + 2 2^) {4.m + 2 x^). 6. (:r-a)(2:-5a); {a-5b)(a + Sb); {a^-2x){a^-6x); (5xio+3a2)(5:z;io_4a2). 7. (32/2-5a:^)(2 2/2-5:?;5); (3 a^ + 2ab)(3a^-4:a¥); (a" + 3) (a» - b). 8. (4 a + 6) (4 a - c) ; (2 & - 5 a) (2 c - 5 a) ; (a y 4- i^;) (a.^ + i^); (af-l)(al+|). 9. (2 2:^ + 1) (2 xi 4- 12) ; (2 a^ - 3 ax) (2 a^ + b) ; (x- .Sx^y''){y- .Sx^y"*). 26. (:r4-7/)(2;- y) = x'^ + {y - y) X x -]- (i- y) X (-y) = 2^2 — ?/2_ Xn which a; and y represent any two numbers. Hence, in general, To find the Product of the Sum and Difference of two Numbers. Take the difference of their squares. ALGEBRAIC MULTIPLICATION. 49 Examples. Find the product of (2 a"* -f- 3 b-") (2 a"» - 3 &-*) ; (8;)* 4- ll2*)(8;>*- Hz*). Solution. (2a« + 36-") X (2a" - 3 ft-*) is the square of 2 a*", or 4 a* •", minus the square of 3 6"", or 9 6-**. Therefore, (2 a"» + 3 6-*) (2 a" - 3 6-*) = 4 o* »» - 9 6- «*. Similarly, (8/)* 4-112*) (8p* - 11 z*) = 64/)« - 121 z. Exercise 18. Write by inspection the product of the following : 1. (2x-^'Sy)(2x^Sy); (x -{- 2ij){x-2y); (5 + 3 a;) (5-3a:); (5a:+ 11) (5 2; -11). 2. {2x-h l)(2a;-l); (2x+ 5){2x-b); (5xy + 3) (5 a; y — 3) ; (c -f a) (c — a). 3. (c2 + a2) (c2 - a2) ; (m n -H 1) (wi n - 1) ; (« y^ 4. j) {af --}))■ (a2r2+ l)(a2a:2_ 1) 4 (a:* + 7/) (ar* - /); (1 - pq) (1 -f- pq)- {m - n) (m^-n)\ (a"* -fa") (a* -a"). 5. {bxr^^^y^{oxy-^-V4.f); {h^+^f)(^2?-?>f)) {2!^-Zx){j^+ 3a;). G. (2aa:4- fey)(2aa;-6y); (m"' + 7i-») (m"'- 7^-»); (10 a-" - 13 6—) (10 a— + 13 6—). 7. (mi + rA) (mi - ni) ; (4 ai- 20a;io) (4ai + 20 a:iO) ; (ai-6-f)(«i + 6-i). 8. (11 a:i + 30 ?/*) (11 a:* - 30 y*) ; (15 a2 6^ - 16 a* 6^) 9. (i«6-2+56-Ja:-i)(Ja6-2-i6-ia;-i); (a+6)(a-6) 50 ELEMENTS OF ALGEBRA. 10. (ah+l)(ab~l)(aH^-\-l); (2a'"+ 4a")(2 a"'-4a") (4^2"^+ 16^2"). 11. (5 a^ + 6&2) (5 a^ - 6h^) (25d^ + 366*) ; {a-^ + aH^) 12. (rc-l + x-Uj) (x-^ - x-'y} (^x-^ + x'' f} ; (f cr- + i If) (f c- - |-6«) (f f c-'^- + }f 62"). Queries. In finding the product of monomials, why add expo- nents of like factors ? What is the product of a^ and a^ ? Prove it. Why is the product of an even number of negative factors positive ? How prove (1) and (2) Art. 25 % Miscellaneous Exercise 19. Multiply : 1. 2 ^2" - a" + 3 by 2 a2- + a" - 3 ; 5 + 2 a;2« + 3^ by4^«-3^2a 2. ft^ + 2 a^" - 3 by 5 - J a" + 2 ^2* ; J a;i - 5 + 8 a;t by \x^ + lx~'^. 3. 3 ^1 _ a - a^ by f a^ + a"! - 6 a-i ; 2 a*""^ &-"' + a-'^ 6^ by 3 a'" ?^'^ - ««^ Ir^K 4. ^x''if—'ix-''y-^ by 4 ^«?/ + 5^2a^26. ^t" + «-t" by ai" + «~^". 5. .3 ft* - .02 ft36 + 1.3 «2^2 + .5 a2,3_ 1 2 h^ by .3^2 - .5 a & - .6 2>2. 6. 1 - 2 ^^ - 2 ^i by 1 — .T6 ; al - 8 «-t + 4 a-^ - a^ by ia~^ -\- a -\- I a-\ 7. 2x^-x^-3x-^ by 2 x'^ - 3 x~^ - x-^ ; a" - 1 + ft- " by a^ + ft~i ALGEBRAIC MULTll'LICATlON. 51 8. ^■""■'■^ - x-" + ' - X-+ x"-' by 0;" + '^ - jJ" - ^; + 1 ; a,'»+3a;"-*-2a,"-^ by 2 .x' + ' + it:- + » - 3 x*. lU. 3^- 2a;'" + ' - 5a;'" + =»+af* + ^ by 3a;"-3 + 2 3;"-* 11. a:" + ^- 3x-+*+ x*+''- 2ic- + * by 2a;'-"+ Sic^' — 12. 5a;"-V-*-'-2a;--^/-^'-u;**-'/+' by 3 a;*+ */"' 13. //«.' + ' — 3m'*?i+m'-^7i2—?/i'-^t^ by m*"-^— 3m-?i 14. 2.«;"+V"'4-3a=*'"'/'"^-a;"+Y'"' + 4x«+V'"' by 2./;■+y-'**'-3^-•^Y-'' + .tV"'' + 4x"-y-^ 15. x-+*y-* + x'-^*if-' - 2a;- + V~" - 4a:- + 'y-* + 4a;»— y^"- 16. (y+ a;-)(y- .»:-'") ; (i^^r -f •^"VXi^rHf^c-/^). 17. (x'' + y~)ra:'*-y-); (a:i - 5) (xi ^ 4), (7 x ^ 3y-^) (7x+3rO. 18. (4 X' - 5 x-^) (4 a; + 3 x-^) ; (f A b'^ - ^^ a^I)^) (|cU-t + ^a^65); (a- 4- 7 + 3a-'')(a~ + 7-3a-''). 52 ELEMENTS OF ALGEBRA. CHAPTEE V. INVOLUTION. 27. Involution is the operation of raising an expression to any required power. hivokition may always be eflected by taking the expres- sion, as a factor, a number of times equal to the exponent of the required power. It is evident from the law of signs that even powers of any number are positive ; and Ihat odd powers of a number have the same sign as the number itself. Thus, (— m* uY = (— m* u) X (— m^ n) (- m* 7i8)8 = (- m*?i8) X (- w* n^) X (- m* n^) = -m< +4 + 4^8 + 8-1-8 =:-ml2u». (- 3 m8 ny = (- 3 m^ n) X (- 3 m^ v) X (- 3 w^n) X (- 3 m^ w) ^ + 31 + 1 + 1 + 1^8 + 3 + 3+3^1 + 1 + 1+1 = 4. 81 mi'-^n*. (a** 6<^)~ = a*^ b" X oT^ ¥ X a"^ b*" X ton factors = (a*" X a"* X a"* X .... to n factors) X (6^^ X 6*^ X 6*^ X ... . to n factors) /'Qm+m + m+.. .. ton terni8\ y^ /'^c + c + c+.... ton termsA — qM X n ^ ^c x n — Qmn ^cn^ where c, m, and » are positive integers f a and 6 may be integral or fractional, positive or negative. Similarly, (a"*6<^# joO" = o*«"6<^"cZ*« .... ;?♦•". Hence, in general. INVOLUTION. 63 To Baise a Monomial to any Power. Multiply the exponent of each factor by tlie exponent of the required power y and take the product of the resulting factors. Give to every even poiver the positive siyyi, ami to every odd power the sign of the monomial itsdf. Notes : 1. Since, aw - - 1 X a"», the nth power of - a™ -» (- 1 X a"*)" — (— 1)* X «"•». Or we may write ± a'"»», for the nth power of - a»»», where the positive or negative sign is to be prefixed, depeudiug upon the value of n whether an even or odd integer, in being positive and integral. 2. Any power of a fraction is found by taking the required power of each of Uliistrations. (-3x'»/)« = -3''<«a:«'<V'' =-27x«y« Exercise 20. Write the results of the following : 1. (4aH*)2; (3a668)3; (2 a; V)^ (^a^V^c^f; (.1 a'»6-)^ 2. {1 a^V^f) (llaJ2c8(fiO)2. (-3cx3/2«)8; {ZaH'^iff; {5abcc^y^z^y. 3. (-2a2)c2^7/*)8; (-abcdxy^- (-a^l^cf; (-^^O*; {SaPc^; {-2aV^f. 4. (lXa<68c2^)n; (-2a2"6"»)^ {-Zxyzf\ (x^^^^^ "-)•"; 5. (-2a''66«.)8. (^•)«. (a«)a. (ft ft -ic- 2)6; (m"7i— )•"•; (-2)8; (-«)«"; (-ir. 54 ELEMENTS OF ALGEBRA. 6. n(2 a b'^ c n "i)*; n {n^ m^*"")"; m (m" a"^)"; 7. 2 a(- 3m7i3a;*)3; m(- 3 aiojs^^e^*)*; a>^ (3 a-^J-^; a(a«-')«. 8. C2x^y\zhy\ (-ra;V"^T; (-3ain-"c)"; (_ 3«-''6't^iyA)6. Affect the following with the exponent 7 ; that is, raise each to the 7th power. 10. {-x''y'^f-{ahh^f;{^^a})xy\ [{-x'^yf ', {-T^mTf. Write the nth. powers of: 11. mia-'^ciyix-yy; (a-3 d)^ '' (x-yf ; 3 (a-b-\-c-\-d) {a — xy. 12. a&c(a-6T(^ + 2/ + ^T; a''{x-y-^zf''\x-y'^f''\ 28. It may be shown by actual multiplication that : {a + hy =a-H6H2a&; (a-hy =a^+h^-2ab; (a + b-\-cy =a2-}-62_,_c2+2a6 + 2ac + 26c; (a-b-cy =a^+b-^-\-c^-2ab-2ac + 2bc; {a-\-b + c-\-dy=a^+b^+c^-\-d^+2ab-]-2ac + 2ad + 2bc+2bd-\-2cd; etc. etc. etc. In each of the above products, observe that the square consists of : I. The sum of the squares of the several terms of the given expression. II. Tvnce the algebraic product of the several terms taken two and two. INVOLUTION, 55 These laws hold good for the square of all expressions, whatever be the number of terms. Hence, in general. To Square any Polynomial Add together the squares of the several tertns aiid twice the algebraic prodtict of every tvH) terms. Example 1. Square 3 a» — 4 x*. Solution. The squares of the terms are 9 a' and 16 x^®. Twice the algebraic product of the terms is — 24 a^ x*. Therefore, (3 a» - 4 x»)« = 9 a« + 16 x" - 24 a«x». Example 2. Square 2 x' - 3 x* - 1. Solution. The squares of the terms are 4 x*, 9 x*, and 1. Twice the algebraic product of the first term and each of the other two terms gives the products — 12 x* and — 4 x*. Twice the product of the second and third terms is 6 x^. Therefore, (2x»-3x2- l)a = 4x»+ 9x* -I- 1 - 12x»-4x« + 6x«. niuatrations. (2 a"» - 3 x-»)« = (2a "•)2 -|- (- 3 x— )«+ 2 (2 a"») X (- 3 x^") = 4 a*" 4- 9 r-«" - 12 a"» x"*. (x-V-Ky-*+|y'-iy)'=(^V)"+(-iar-tr*)*+(|y^+(-iyy + 2(x-V) X (- ix"y-«) + 2(r-«y'') X(§y'H-2(x-V)X(-iy) + 2(-ix"y-«) X(|y«) + 2(-ix-ir*)X(-iy)+2(fy») x(-Jy) + |x-«y*+t-|r-«j/*+»-§x"y + ix*y-'» 56 ELEMENTS OF ALGEBRA. Exercise 21. Square, by inspection, the following : 1. x + 2; m + 5; n-\-7; a — 10; 2x -\- Sy; a + Sb; a — Sb; 2x — Sy. 2. X + 5y; 3x — 5y; 2a + ab; 5x — Sxy; 5abc — c; xy-^-2y^; a™ + 3 6"". S.2x + Sa^; xy + x'^; 3 a-2 -f 5 a^-^; 1-x; 1 — cy; m — 1; ab"^ — I ; -J a"— .05. 4. 1 a 6-2 + |6-ia;-i; fff-r^'^ | a-« _ 2 j-»«; ic-f 3/-f + J; .0002a;'" + .005/. 5. I m^ n^p^ — ^mrf; xy + yz-hxz; 2x^ ■]- Sx — 1; x^-2x+l; x^ + 2x-4. 6. 2a^—x + S; a^—5x—2; x^—2xy + y^; 4:n^+m^n—7i^; x^ - 3x 4- 2. 7. xy — 2n + 1; m — n — p — q; a^ — 2a^ -\- 2x — 3; 1 + X -}- x'^ + x^; x+Sy+2a — b. 8. 2a^-Sa^-x + 3; x - 2y - Zz + 2n\ wT ^- tT 9. ^a-2b-V\c',xf-y- + \a-\b;la^-x + l', 10. l^lx-\x',la^-\x-\;\ar-\a-+\xy', 2 a;t + 5 a:i + 7. 11. '^x^-2x^ + \x^-x-^; m^"-f a;i«2/-t*-|^'^-3; 2i-3i. 29. Any Power of a Binomial. It may be shown by actual multiplication that: INVOLUTION. 57 (a + 6)3 = a3 + 3 an + 3 a?;^ + 68. (a - 6)3 = a8 - 3 a26 + 3 a 6^ - JS; (a + 6)* = a* + 4a36 4- 6 a^l^ + 4:ah^ -^ 6*; (a - 6)* = a* - 4 a36 + 6 a2 62 - 4 a68 + 6*; (,f + 6)'^ = a^+ 5a*6+ 10 a8 62 + 10 a263 4- 5a6*+6«; (,i -6)'^ = a6-5a*6+ 10a362- 10a263+ 5a6*-6fi; and 80 on. In each of the above products we obsei-ve tlie following laws: I. The number of terms is one more than the exponent of the binomial. II. If both terms of the bin^omial are positive, all the terms are positive, III. If the second term of the binxrmial is negative, the odd tei-ms, in the product, are positive, and the even terms negative. IV. TTie first and the last terms of the product are respec- tively the first aiul the last terms of the binoinial raised to the power to which tlie binomial is to be raised. V. The exponent of tlie first tei^m of the binomial, in the second term of the product, is one less than the exponent of the binomial, and in each succeeding term it decreases by one. The exponent of the second term of tlie binomial, in the second term, of the product, is OTie, and in each succeeding term it increases by one. Thus, omitting coefficients, (a + 6)« = a« + a^b + a*62 + aS^s ^ «2 j4 j^ ab^ ^ 6« VI. The coefficient of the first and the last term is one, that of the second term is the exponent of the binomial. 58 ELEMENTS OF ALGEBRA. The coefficient of any term, multiplied by the exponent of the first term of the binomial in that term, and divided by the number of the term, will be the coefficient of the next term. Notes : 1. The sum of the exponents in any term of the expansion is the same, and is equal to the exponent of the binomial . 2. The coeflScients of terms equally distant from the first terra and the last term of the expansion are equal. Thus, we may write out the coefficients of the last half of the expansion from the first half. If one or both of the terms of the binomial have more than one literal factor, or a coefficient or exponent other than 1, or if either of them is numerical, enclose it in parentheses before applying the principles. Thus, Example 1. Expand (2x'^-5a^xy Process. (2 a:8 - 5 a2 a;)* = [ (2 a;3) - (5 a2 x) ]4 = {2x^y~4{2x^f{5a^x)-\-6{2xy{5a^x)^-4{2x^){5a^xy + (5a^xy = 24a;i2_4 X 2^x^ X 5a^x-\-6X 2^x^ X b^a^x^- 4 X 2a:8 X5^a^x^ + 5^a^x^ = 16a;i2_4X8a:«X 5a^x-{-6 X 4x«X 25a*a;2-4x 2x8 X 125a^x^+626a^x^ =z 16a;i2_160a2a;io + 600a4a:8_1000a6x«+625ft8a:* Explanation. In the expansion the odd terms will be positive, and the even terms negative. The first term is (2 x^y, and the fifth or last is (5 a^xy. The exponent of (2 x^). is 4, and in each succeed- ing term it decreases by 1. The exponent of (5 a^x) is 1, and in each succeeding term it increases by 1. The coefficient of the second term is 4. For the second term we take the product of 4, (2 x^)^, and (5 a^x). To find the coefficient of the third term, we multiply the coefficient of the second term 4 by 3 (the exponent of (2 x^) in that term), and divide the product by 2 (the number of the term), and have 6. Hence, the third term is 6 (2 x^y (5 a^x)''^. The coefficient of the fourth term is found by multiplying 6 (the coefficient of the third term) by 2 (the exponent of (2 x^) in the third term), and INVOLUTION. 59 dividing the product by 3 (the number of the term). Hence, the fourth term is 4 (2 x*) (5 a* a:)*. Performing operations indicated, we have the required result. Example 2. Raise 1 — § x" to the fifth power. Process. (l-§x")» = (l)»-5(l)*(|x«) + 10(l)»(§x»)«-l0(l)«(fz»)» + 5(l)(fxn)* = 1»-5X l*x|a*+10X l«x|x2--10X l«X^x»« + 5X 1 Exercise 22. Expand and simplify the following expressions : 1. (a - 6)7; {a 4- x)^; {a^ ~ ac)*; (a^ - 4)^; (2 + a)*; («-l)^ (1 -aY; (2a-Sby. 2. (xh - 3)* ; (ax-S x^f ; {x - 3)^ (2 a^z + 3 62^)8; (2ax-\- Sbyy. 3. (« + 2)e; (a-2)«; (2-Ja)*; Cja-36)*; (Ja + }6)*; (a + 6)W. 4. (ai - 2 - a-^y ; [(2: + y)H {x - yff\ (1 + a + a^)^ - (1 - a + 2 a2)'f 5. (a + 2i)*-(a-26)*; (3 - 2a + a2)2 - (2 - a)*; (3i + 5i)2 _ (2i - 3i)2 Queries. How prove (— m)* = ± m", according to the value of n, whether an even or odd integer ? How prove the method for squaring any polynomial? How prove the laws for raising a bino- mial to any power? 60 ELEMENTS OF ALGEBRA. CHAPTER VI. ALGEBRAIC DIVISION. 30. Division is the inverse of multiplication, and is the operation of finding the other factor, when a product and one of its factors are given. The product is now called the Dividend, the given factor is the Divisor, and the required factor is the Quotient. Thus, since a^ X a^ = a^ .*. a^ ^a^ = a^-^ since a-^Xa^^a^, .*. a^-^a^ = a-^\ since a^ X a-* = a, .-. a-^a-* — a^; since a"*-" X a" = a"*, .*. a"» -r- a" = a"»-"; since «"»+" X a-" = a"*, .*. a™ -^ a-" = »*"+" ; since Sa^ft* x 2a-26 = 6a6^ .-. Qah^ ^^a-H = ^a^b^; since 9a-362 x 3a*65 = 27a6^ ... ^1 aU' -^Za^h^ = ^a-H^; since 5 a*6"~^ X 4a~*6^ = 20aH*, .•. 20a^h^^Aa~^h^ = baH~^ ; etc. Hence, in general, To Divide a Monomial by a Monomial. To the quotient of the numerical coefficients annex the literal factors, each taken 'with an exponent obtained hy subtracting its exponent in the divisor from its exponent in the dividend. Illustrations. a^h^c^m^-^ a^b^c^m^ ap-^b'^-^c^-'^m^-'^ =abc*m. 63a-26V5^ 7«-36c'» = 9a-2+3i2-ic5-4 =()abc. 1^ 2A2 . «^ 5a2i2-i ^a^b ,. ^ „ I5a^b^-^6bc = — = (Art. 2). 2c 2 c ^ ^ ALGEBRAIC DIVISION. 61 Exercise 23. Divide : 1. 3a362 by ab] I6a*i^ by 3aH^; 20a^}^c^ by 5a6V; Smi by 5 m^. 2. tT^ by ?i~i^; a* by a*"^; a^j-s^n y^y ^35-2^2. ^m+i. by a"* — ; 2'+' by 2'-'. 3. 15 a-t 6-i a^ by 9 a-2 6"! ic^. ^ ^i fti by f (A ji ; 21a*m2ic' by Tama:*. 4. 24a"j9'" by 3a>"; 36a'"mV^ by 9amyri*; «*'+y-* by a^/. 5. (x - y)6 by (a; - y^; (a - c)*+8 by {a - c)*-i ; |fe*^^•- by f 6/H^. 6. (6a3 62,; X iSftSJV) by (S^a^'c^ x 2a*c8); a*"' by a*"; (2 77i7i")2* by (2mn«j^ 31. Only a positive number, + a, when multiplied by + 6, can give the positive product +a6. Therefore, +ab divided by +6 gives the quotient + a. Thus, since aX6 = + a6, .•. 4-a6-f + 6 = -f-a; since aX-6 = — a6, .*. — a6^ — 6 = -|-a; , since — aX6 = — a6, .*. — a6-^ + 6 = — a; since — aX— 6 = 4-a6, .*. -{- ab-. — b = — a. Hence, in general, Law of Sig^. If the dividend and the divisor have the same siguy the quotient is positive. If they have opposite signSy the quotient is negative. 62 ELEMENTS OF ALGEBRA. Example. Divide 12a"» by — 4 a". Solution. Since there is a factor 4 in the divisor, there must be a factor 3 in the quotient, in order to give a product of 12 in the divi- dend. Since there are m factors of a in the dividend, and n in the divisor, there must be m — n factors of a in the quotient, in order to give a product of a"» in the dividend. Hence, 120'" -f - 4 a*^ = — 3 a"*-", because only a negative number, — Sa"*"", when multiplied by —4a" can give the positive product, 12 a*". Illustrations. - ISa^mHS-f 3a2m*62 - - b a^-^b^-^m^-'^ = ~5a^bm^. - 5 x^^y^z^ i- -10 x^y^z^ = + ^ cc^o-s ^^8-5^6-3 ^ ^ ^x^y-^z\ ia'{a-by{x + y)^^ -4:a{a-b)^{x + yy= -|a"-^(a-6) (x-t-i/)"*-". Exercise 24. Divide : f. 6^ by 3^; -20aH^cJ by lOahc; 35a^^hy-7a^; -laHc^ by -7aHc^. 2. 27ax* by -9^4. _|a6^,6c6 by ^aHc'^; .^aH^^c^^ by fa^^iici*; 12^2"?/ 2 by -f^j''?/. 3. Z\7n?n^3iP- by -2i m-i7i-3^-2; - 5| m-^j^-iyio by *lj2ga2m3x-4?/; 3.2Jrt-s^?/S by 2.Qt2\a-^xy\ 4. .O^a^wiV^* t)y -.0|a2^2/2^3. _9.3m3«^2-^«-32/§c by .3m3«+ix'*-4 2/K 5. .^x'^ifhy-ix'y-'^'^- -J(a fe)3c8 by .6(a-&)2ci0; ~ .3ai'"ft^ by -.2 a" 6". ALGEBRAIC DIVISION. 63 6. - .375 xi ?/U-^^ - y^^ by - i^ oc^ y (xi - i/l)l . 8m-Si-^r-02y7 by 9 m ^ ,. "- x-^ y- \ 7. -1.2aiO(jc-y)"r» hy .^a^{x-yfz^'^\ m-^n^x-yY (y-zY by w 2«n2''(a;- 7/) '•(?/- 2)i^. Simplify the following, that is, perform the indicated operations : X -~.5a2''62«c-2'. 9. {a-H*-^2ab) x -2a2fe-2 x (- .Gaifti -=- - .3aUi). 10. (.3a-'"6-'*c-''H- .03a"'6-c'') -r 1 j «-3'"6-3»c-3''A:. 11. (4§«-»6(ijc-2-f-lia-U-3rf*) X [6 a2c- irf3 ^ ^ (84 ^8 j8 c -f- 7 a* b^ c^)]. 12. (ic""^'* X a-U-i.i-"-") x (rtUa^^-^^^i -!-6ir*'"7/-t). 13. (-Ua^H^c-^^-7aH*c-^) -=- (28 a-^iV ^ - 4 a-'^b-^c-"). 14. (1.7«-i6-^cijc2-M.l«-2i-ia:8)x (a"'i*c-6^al62c3). 32. Since (a •^- b) m = a m + b m, .-. (am + bm) -r m = a + b. Since (a - b) m =■ am — bm, . • . (a m - 6 w) f m = a - ft. Since (iy-2y«2-3x»ir') X -3xy»= -3x«y< + 6xi/*2 + 9x*^, .-. (-3x»y*4-6xy»2 + 9x<y)T -3xi/» = 2r/-2ya2-3x»y-». Hence, in general, To Divide a Polynomial by a Monomial. Divide each term of the dividend by the divisor, ami add the results. 64 ELEMENTS OF ALGEBRA. Exercise 25. Divide : 1. 2a?+ ^a^y ~ 8a^y^ by 2a^- 21 m^n^ - 1 m^n^ — 14 myi + 63 by 7 mn. 2. a^hc -- a? b^ c2 - a^ W' c^ + a^ b^ c^ by a^bc; 42 a^ — 1.1 a;2 + 28 ^ by .7 x, 3. 28a3 + 9a2-21a + 35 by 7a; 4: a^ b^ - 16 aH^ + 4:aH^ by - 4 aH^. 4. 66t262c3 _ 48^264^2 + 36^2^2^! _ 20abc^ by 4a&ca 5. 2.4 m27i2 — .8 mht^ — 2.4 m ?i2 + 4:m^n^ by .8 m n ; icf — x^ y^ by ici. 6. -"^a^^^ab-^ac by -1.5a; .5 m57i2_ 3^3^* ^y — 1.5 m^v?. 7. - 72 o^ c2 - 48 a} ^10 + 32 ^2 c^ by 16 a^ c^; 3.6 n* — 4.8 rtf by 4 n\ . 8. 11^2^3^ 3 a; 71 -2.4 2/2 by .'^xy\ .09 7^1*- 2.4 m^ 71 + 4.8 «i5 by .03 wi* 9. - ft"* + 2a-"' - 3a" by - a^\ m"+i- 7/i"+2 + 77i'*+3 — m"+* by m?. 10. 2.1 a 2:2 y"* 4- I.4a3;:c4^"- 2.8a5^2^P by - ,1 axy"". 11. a'"53_^-+ij2_|_^n-2j by -ab; -2a^a^-j-S.5a^x* by 2.3^ a^x, ALGEBRAIC DIVISION. 65 12. 2.25 a^x - .0625 aire - .375 a ex by .'67^ ax; llxi-33a;* by 11 x*. 13. 72 wt - 60 mi ni + 12mi ni - 6mT^ni by 24 mi. 14. 36 (x - yf - 27 (x-y)3 + lS{x-y) by 9(a:-y). 15. -123ry*2'S0sf''-^^y^z+10SaP^y2f-^^ by -6^y*^ 16. m" (x — y)« — 7;ia(a; — t/)" by m* (a; — y)*. 17. {x-hyY(x-yy-\-(x+yy{x-yy by (x-^yy{x-y)\ 18. -2.5m2+1.6m7i+3.3m by -.83m; a-^i-ai^ift^+a ^ by a-w. 33. It may be shown by actual multiplication that : (TO+n+/>) (x+y-f-z) = ma:+TOy+m2 + nx + nT/ + nz+;)a;+/>y+/)2. .♦. (nu:^-my+'nu-\-nx-\rny-]rnz-\-px+py-\-pz)^{x + ]i + z) = m-\-n-\-p. The division is performed as follows : Separate the dividend into the three parts mx + my-fmz, n X 4- n y + n 2, and px + py-\-pz. The first term of the (quotient, m, is found by dividing m x, the first term of the dividend, by x, the first term of the <U visor ; multiplying the entire divisor by m will produce the Jirst part of the dividend. The second term n of the quotient is found by dividing the first term of the second part of the dividend by the first term of the divisor ; multiplying the entire di- visor by n will produce the second part of the dividend. The third term p of the quotient is found by dividing the first term of the third part of the dividend by the first term h{ the divisor ; multiplying the entire divisor by p will produce the third part of the dividend. The work is conveniently arranged as follows ; 5 ( ^MV£RS(Ty\ 66 ELEMENTS OF ALGEBRA. g + g ■v..^ w M f M N ^i fts a, ^1 4- + 4- + ?=^ 5>i ^ 5ss a^ a, a, a^ + + + + H « H H a. ^H ^i a, 4- + M l^ ?\i $ 8 s + + + ?s» 5rj ?s^ S S S + + + « H H s s S 4- N N fc s + + >i a>i S g + + « « 1. ^ N ^*" + ^- >. -g •+3 d ■^ d quotie O 'S -l a ^^"^ "o 1 f 1 1 -•^ ^ o •S ^, ^3 o ^ 05 '^ ^ ^ ^ '^u c ^ si i=l d a ^ rt 13 3 f-i rt ;-• F-i o o d O w Tl •iH 'eS ^ % _> '> 'S § 'S OJ ^ O) (B 2h O) ^^ _t- J c "3 rt d O) cj <u (u %-i ■^ V-( n:J ^_i o B o d c3 o -u S -M -u * o o "73 o :=i |3 rt d -ij g ^ n:) o o Q o &4 ;h PU S fl^ m PU + + CO I I ^ I I 7 I CO I + + CM H H 1 CO 1 1 1 1— 1 4- H (M 4- 1i CO 1 1 1* 1 1 1 1 « 1 1 CO 1 1 ^ 1 1 1 1 1 (?q -t CO CO + 1 4- 4- "« TO CO CO + + Tj ^ t, (M (M 1 i 1. 1. 1 1 . CM 1 1 fj" o .2 a> f ^ J 1 m" *> ^ d 0) .^ d •5 1 "-I3 d 0) d ■^3 % ^ ^ +3 S S i 1 (N CO •1-1 1^ ptH 1 OQ 1 ALGEBRAIC DIVISION. 67 Explanation Dividing the first term of the dividend by the first term of the divisor, we have a:*, the first term of the quotient. Now as we are to find how many times x* — 3a:^-j-2z+lis contained in the dividend, and have found that it is contained a:* times, we may take X* time.s the divisor out of the dividend, and then proceed to find how many times the divisor is contained in the i-emainder of the divi- dend. Dividing the first term of the remainder by the first term of the divisor, we have — 2 a:, the second term of the quotient. Simi- larly, we find the third term of the quotient. Hence, the quotient is x« - 2 X - 2. Notes: 1. Algebraic division is strictly analogous to *Mong division*' in Arithmetic. The arrangement of the terms corresponding to the order of suc- cession of the thousands, hundreds, tens, units, etc., and the processes for both are exactly the same. 2. It is convenient to arrange both dividend and divisor according to poioers of the same letter ascending or descending. 3. It may happen the division cannot he exactly performed ; we then alge- biaicaUy add to the quotient the fraction whose numerator is the remainder, and whose denominator is the divisor. Thus, if we divide x"^ — 2xy —i^ by X — y, we shall obtain x — y in the quotient, and there vnll be a remainder — 2ya. Hence, (xS - 2xy - y^) -r (x - y) = x - y - ~~. ■ X y Example 2. Divide a* + 6' -I- c» - 3 a 6 c by a + 6 -f- c. Arranging acconling to the descending powers of a, we have: Process. Dirisor. Diridend. Quotient a-f6+c)a» -3a6c-|-6»H-c»(a*-<26-ac a* times the divisor, a»-t-a^fe-fa^ First remainder, -aV)-a*c -3a6c+6»4-c» — ab times the divisor, —aV* — ab^ — abc Second remainder, -a^(H-ab^ -2a^c+6»+c» — ac times the divisor, — q^c —ac^— ahc Third remainder, ah^^ac^ abc-{-l^-\-c* b^ times the divisor, ab^ ■ -\-b^-\-b^c Fourth remainder, ac*- abc-b^c+c* c* times the divi.sor, ac' -^bc^-c* Fifth and last remainder, -abc-b^c-bc^ — be times the divisor, —abc—b^c-bc^ To verify the work, multiply the quotient by the divisor. 68 ELEMENTS OF ALGEBRA. Example 3. Divide i^ xy^ + \x^ + ^y^ by ^y + ^x. Process. ^x + ^y)\x^ +^xy^+^y^{j^x^-^xy + {y Divisor X ^x\ ^x^ + ^x^y First remainder, — i ^^2/ + -^ ^ 2/^ + i^y^ Divisor X — ^xy, — ^x'^y — ^ x y^ Second and last remainder, i ^ ^^ + iV 2/* Divisor X \ y% i ^ J/" + i^ y^ - Hence, in general, To Divide a Polynomial by a Poljmomial. Divide tJie first term of the dividend hy the first term of the divisor for the first term of the quotient; multiply the entire divisor hy this term, and subtract the product from the dividend. Divide as before, and repeat the process until the work is completed. Exercise 26. Divide : 1. 14 x^ + 45 ^2/ + "^8 x^i/ + ^bxf +14.y^ hy 2x^ + 5 xy + 7 y\ 2. x'^ — 2x^y+2x^y^ — xy^ by x — y; a^ — 2ah^-^h^ hj a-b. ^ 3. f-5y^ + 9 2/-6y^-y+2 by y^ - 3 y + 2 ; y^-1 hy y-1. 4. x"^ + xy + 2 xz—2y^+ 7 yz — 3 z^ hy x — y +3z; c^ —b^ by a — h. 5. 2^2/+ 36?/ + 10fe;:c + 15 62 by 2/ + 5&; a6 + a^b hy a -\-b. 6. .125ic3-2.25a;22^4. \Z,^ xy^ -27 y'^ by .^x-3y, 7. ?/-62/^-2^ + 54^-3a;2?/ by 2x-y\ x^-y^ by x + y. ALGEBRAIC DIVISION. 69 8. a^t/^—a^ — y^+lhyxy + x-\-i/+l;4:i/ + 4:y -y8 by 3y + 2i/2 4-2. 9. s^+i^ — z^-{-3xyzhyx + y — z; x—y by x^ — yi. 10. 3^y^-\-2xi/^Z'-a^z^-{-i/^z^ by xy + xz + yz; x^ — y^ hy xi — yi. 11. 12x*-26a^y-Sx^f+10xf-Sy^hySx^ -2xy-\-f. 12. a^^'f^-Zxy-\hyx-\-y-\) 5V ^ " iV ^ + iV ^ - S*I by J X- - i. 13. 12 ;c« ?/9 - 14 x* 2/^ + 6 a;2 ^9 _ ^9 by 2x^y^-f) a^ — y^ by a;« — y^. 14. a^ 6 - a ^ by a3 + 63 + a ^2 + ^2 J . ^ ^ ^4 _ -^ — x~^ by a; — x~^. 15. a^ + :r* y + a:^ 2/* + i^ ?/^ 4- a: ?/* + 7^5 by a;^ + 7/^ ; al — 6f by a^ — fci 16. Ja3 + Y«^-l-25a + 2.25 by Ja + 3; .Ibx^y^ + .048 a:^ by .2 a;^ ^ 5^,^ 17. at-a2-4at + Ga-2ai by J-4«i + 2; a:S_y5 by a; — ;/. 18. a;8 + 7^ + 23 4. 3 ^3y ^. 3 aj^J by a; + y + 2 ; .5 a;8 + a.^+ .375 a: + 75 by J a; + 1. 19. x^-\-^y^^-^-^xyzhy 3^-\-^f^7?-xz-2xy -2yz. 20. ^g a:* - I a,-3 - J r^ + I a: + Jg^ by 1.5 a:^ _ 3, _ | 21. ofi — 1^ hy a^ + Qc^ y -\- X y^ ■\- ij^ , a^ — 7^ by a^ + a; y + 2/2. 70 ELEMENTS OF ALGEBRA. 22. 10 a^--27aH + 34:a^I^-lSah^-8b^hj 5 a^-6ab - 2R 23. 36x^+^i/+.25-4:X7j-6x + ^y by Qx-^-.d. 24 ai2 + 2 aH^ + Z^^^ by «* + 2 ^2 ^2 + ^,4. ^6_2,6 ^y a^-2a^b + 2ah^-h\ 25. 2 ^^" — 6 iz2« y- + 6 rr"2/2" - 2 ?/3« ^y 2^ — 2/«; ^» + 2/3" by it" + ?/. 26. i6'2«— 7/2'" + 2 7/"*^'- s2' by ^''+ ?/"• — ;3'; 32^-22^ by 3^ - 2". 27. ifX^-lil-xi/^ by f^-.752/; a:-t"'-3aj-i"'2/-i'' + 2 y-h by a- ^"^ — 2/-i". 28. ?/2a;2m^2 7/22:'"+"+ 2 2/r a?"* + 22a;2n + 2ra?";2 + r2 by 3/ a:*" + 2; ^" + 7\ 29. a;~i + 2x~^y~i + 2/"^ by a?~^ + y~^; x^ + ?/* by a?2 + 22 a? 2/ + 2/2, 30. x~'^ — y'~'^-\-2y~^z~^ — z~'^ by a;~i + 2/~^~^~^j a?* — 3 2/* by x — y. 34. There are special methods for finding the quotient of binomials, hy inspection, which are of importance on ac- count of their frequent occurrence in algebraic operations. Thus, It may be shown by actual division that : a—b ^a—b ^^—^ = a^+as 6+a2 h^+ab^+b^ ; ^"—^ = a^ + a'^b + a%'^ + 02^8 +ab^+b^) a—b a—b and so on. Hence, in general, it will be found that, ALGEBRAIC DIVISION. 71 The difference of any two equal powers of two numbers is divisible by the difference of the numbers. In each of the above quotients we observe the following laws : I. The number of terms is equal to the exponent of the powers. II. Hie signs are all positive. III. The exponent of a in the first term is one less than the exponent of a in the first term of the dividend, and in each succeeding term it decreases by one {in the last term its exponent is 0, or a. disappears). The exponent of b in the second term is one, and in each succeeding term it increases by one {in the last term its expo- nent is one less than the exponent of b in the dividend). IV. The first term is found by dividing the first ter^n of the dividend by tJie first term of the divisor. V. To find each succeeding term, divide the preceding term by the first term of the divisor, and multiply the restUt by the second term of the divisor regardless of sign. Example. Divide 1 — w* by 1 — n. Solution. Dividing 1, the first term of the dividend, by 1, the first term of the divisor, we get 1 for the first term of the quotient Now divide the first term of the quotient by the first term of the divisor, and multiply the result by n, the second term of the divisor (regardless of sign), for the second term, n, of the quotient. Dividing the second term of the quotient by the first term of the divisor, and multiplying the result by n, we have n^ for the third term of the quo- tient. Similarly, we find n*, and n* for the fourth and ffih terms, respectively. .*. (1 — n*) -r (1 — n3 = 1 -|- n -f n* + n* + n*. 72 ELEMENTS OF ALGEBRA. Exercise 27. Divide by inspection : 1. m^ — n^ by m — n\ a^ m^ — h^n^ by am — hn\ m^n^ —1 by mn — 1. 2. l—m^n^a^hjl—mnx-jix yj*— {x zj hy xy — xz; 1 — a''b'' x'^ hj 1 — ahx. In order to apply this principle the terms of the divi- dend must be the same powers of the respective terms of the divisor. It is not necessary that the exponents of the terms of the divisor be 1, nor that they be the same, nor that the exponents of the terms of the dividend be the same. Thus, Example Divide x^^ — y^^ by x^ — y*. iSolution Dividing x^^ by x^, we have x^ for the ^rst term in the quotient. Now divide x^ hj x^ and multiply the result by y*, for the second term, x^ y*, in the quotient. In like manner we find x^y% and y^^ for the third and fourth teims of the quotient. . •. (a;i2 - i/16) -i- (x3 - 2/4) = a;9 + ^6 ?y4 + a:3 2/8 + ^12 So in general x"* — y"^ divides 2:"" — ;v""* {n being any positive integer), since the dividend is the difference be- tween the nVa powers of the terms of the divisor. 3. a^ - W by a^-l^- x^^ - f^ by x^ - y'^ \ x^^ - y^'^ by a::^ — y'^. 4 ai5 - &30 ]3y a3 - &6 . x^m _ ^35n -^^ ^n _ ^n . 2io« - a^"" by 22" - x"^. We may easily apply these principles to examples con- taining coefficients as well as exponents; also to those involving fractional or negative exponents. Thus, ALGEBRAIC DIVISION. 73 Example. Divide 81 a^^ - 16 A** by 3 a« - 2 h\ Solution. Dividing 81 a" by 3 a«, we have 27 a» for the frst term of the quotient. Now divide 27 a* by 3 a* and multiply the result by 2 6*, for the second term, 18 a® 6*, in the quotient. Simi- larly, we find 12 a' 6^*, and 8 h^^ for the <At>d and fourth terms in the quotient. .-. (81a"-16634)^(3a»-26«) = 27a»+18a«6«+12a«6»2+86". If a and h are coefficients, a^a^** ~ 6"^*"" is divisible by ax' — hif^f since the dividend is the difference between ^1 -1 the Tith powers of ax* and ////'". In general, a? "•— y *" na n$ divides x '^ — y "" (n being any positive integer), since a the latter is the difference between the ?ith powers of a; •* and y '. 5. 64ai2_27?i» by 4a^-Sn^; 16 2^ y^^"* - ^ m^z^* by 4a:8y6*_ j^^4 2io» 6. «8 a^S" - 68 2/3« by a2aJJi' _ ^^ ^^^ . 32 ar^o - 243 y^ by 2 0^2 - 3 3/3. 7. x~^ — y~i by a;~i — y~i ; 3^ — y^ by a;^ — yi ; a xi — 6i y by ai x^ — fci yi 35. It may be shown by actual division that : ai—l)i a*-h* — r-j- = a — b; — —r- = a*—a^b-{-ab^—b*; and so on. a+o a + o Hence, in general, it will be found that, The difference of any two equal even powers of two num- bers is divisible by the sum of the numbers. In each of the above quotients we observe the laws are the same as in I. and III., Art. 34 ; also, 74 ELEMENTS OP ALGEBRA. VI. The signs are alternately + and — . Hence, the principle may be applied to different classes of examples as in Art. 34. Thus, in general, If a and l are coefficients, ax^ -{■ h'lf divides a" x"*^ — If y^"^ (n being any even and positive integer ; also m and p may be integral, fractional, or negative), since the divi- dend is the difference between the nth. powers of a x^ and by-. Note, 'rtie difference of the squai'es of two numbers is always divisible by the sum and also by the difference of the numbers. Thus, 06 — 68 jg divisible by aS ± 64. jn general, a^n — IZm [§ divisible by a« ± **" when n and m are integral. This is the converse of Art. 26. Exercise 28. Divide by inspection : 1. 625 a^x^ — 81 ra^n^ by 5ax+ 3 mn; a^ — h^ hy x:^ + b^; x^ — 1 by x + 1. 2. x^ - yi by x^ + y^ ; 256 ir* - 10000 by 4£c + 10^, 3 ^im^yin by a;"* + 2/" ; tV ^^ - -^^^^ 2^^ by J rr^ + .22/1 ; ^10 _ ^,10 ]^,y a, + 6. 4. 729 ai2 _ 64 ^,18 ^y S a^ + 2 b^ ; a^ x^"" - ¥y^"* by a ^-^ + &2 y2 m 5. a""^ — x'"^ by a~^ + a;"^; a^ x~^ — b^y~^ by a a:~3 + h's y~^- • 6. xh - yi"" by 2;i*^ + t/t^"*; 81a-i"a; - Jq &f«^-f'» by 3 a-T5'*a;4 + | b^y'^'^. ALGEBRAIC DIVISION. 75 36. It may be shown by actual division that : j- = a^—ab + b^', j- = a*—a*b+a^b^—ab^-^b*; and so on. a-\-b a + b Hence, in general, it will be found that, The sum of any two equal odd powers of two numbers is divisible by the sum of the numbers. In each of the above quotients we observe that the laws are the same as in Art. 35. Hence, the principle may be applied to all the different classes of examples as in Art. 34. Thus, in general, If a and b are coefficients, aaf-^-hy"^ divides a" a:"' _j_ i^ynm ^^^ beiug ani/ odd and positive integer, also m and p are integral, fractional, or negative), since the dividend is the sum of the nth powers of aaf and by^. Exercise 29. Divide by insj^ection : 1. x" ■\- If by X -\- y \ x~^ + y~^ by a;"^ + 3rM 1024 a:« + 243 y« by 4 a; + 3 y. 2. 128 x^^ + 2187 y" by 2 2:3 + 3 y2 . 243 x^^ + 32 ^o by 3 a:8 4. 2 y2. 3. a;!*" + 2/21*" by 3^^ ■¥ y^^ - A:2i« 4- 7^86- ^y l^^^m^''', a" + 6" by a + 6. 4. wi w- + a; y by wi ?ii + xi yi ; x^ + ?/V by x^ -\~ yi\ ^\-\- y'^ by a;-i + y-K 5. ax\^b^yhya\x\^biiy\', (# -f ( J) by (f )i + (f )i ; a^ + 6^0 by a + 62. Note. Since a« and Ifi are odd powers of a^ and h^, n^ + 6« is divisible by cfl + 62. aW and ft" are the 5th powers of a^ and 62, a 10 + 6W is divisible by a* + 6^. Also, a» and 6* are the third powers of a« and 6«. Therefore, a« -f 6» is divisible by a* + b^. 76 ELEMENTS OF ALGEBRA. 6. a^ + 612 by a* + &^ 2:6 + 1 ]3y x^ +,1 ', x^^ + 1 by a;* + 1 ; a27 + ^27 by a^ + h^. 7. a;io + 2/10 by 2;2 + 2/2 ; a;!^ + 2/^^ by a^ + y^ ; 64: + x^ by 22 4- 2;2 8. 64 x^ + 729 f by 22 a;2 + 9 ^^2 . ^lo + __l^ by x^ + ( J)2; ^24 4. 524 by «§ + &8 ■ 9. ai8 + Z,i8 by a^ + b^ and by a^+P; j^^ x^ + J^ f by i^^ + i2/^- 10. ^36 + J36 by ai2 + ^,12 and by a^ + J*; 729 a;^ + 1 by 9 a:2 + 1. 11. a;*2 + 2/42 by x^ + ^6 and by x^^ + 2/^*. Query. Is it divisible by 2;2 + / ? Why ? a^^ + &18 by a^ + h^ ', ^27 + J21 by a9 + &' ; a^^ + 515"' by a^ + J^. 12. rr^ + 2/^* by o^^^ + ?/i^ and by x^ + /. Query. Is it divisible by 2^2 + 2/^ ? Why ? a^ + &12 by a^ _^ &* ; aH9 + m27n36 by a^h^ + 77i9 7ii2; ^15^25 _^ m^o^iio by a^ 6^ + m^ 7i2 Find an exact divisor and the quotient for each of the following, by inspection : 13. 8 + a^; ^6 - 6^ 8 - ^; a:* - 81; a^^ - h^^; 81 ai2 - 16 h^; a"^- 625 ; a^ - h^ 14. aj20 - 2/15; ^,^5 + ^5. ^12 _ ^12 . ^6 _ 1 . ^-12 _ 2,-12. a^a^^-h^fp- a^-'62) 16 «* - 81. 15. Z2w> - h^; 81 «8 - 16 b^; 1 - y'^ ; a^ x^ + 1000; a* a;* — 1 ; a^ + m^a:^ ; 2^2 7/2 — 81 (x2 16. 32 ^10 - 243 2>i5. cIo.tIO'* - a^^x^^; a^" + ftQ''^ a^x^ + b^y^""; c^ x^^ + 6' 2/^". ALGEBRAIC DIVISION. 77 17. aj-^" + y-e*; 8a^y3 4. 729; fti2 yi2j» _ Ji:^yi2». cS^^Sp _ j8^«^ 2:4 _ 1296; ^^S" - ftio* 18. 128.^21 + 2187^^*; 256a;i2-81 //«; a6»65* + a:S'y^'; 1 + 128 a:!*; «-«'»- d^^-. ^j-ii^-i-l. 19. x^-y-i_aiy-i; a-i^-S+l; i^a^x^'^^^^h-'^f^ g^ x-t"-. 00032 y-^". 20. ii c" + .002432;2/-i; 256 aa;-t~ - .0081 J-f*; Queries. How divide a monomial by a monomial ? Prove it. How prove the method lor dividing a polynomial by a polynomial \ In Art. 35, the sign of the last term ot" the quotient is — , while in Art. 36, the sign ot* the last term of the quotient is 4-. Why is this ? What is the product of a* and a- * ? Prove it. What of a* and or* I Prove it. Miscellaneous Exercise 30. Divide : 1. a3 6-3+ T^a2 6-2 + ^1 by Jafe-a + JJ-i; ari + y-i by x~^ + ir^- 2. a-6+ 5a-H-i+10ft-36-2-f i0a-2 6-3 + bcrH'^ + 6-6 by a-i + b-\ 3. 2 a2 - a^ - 2 rt + 1 by 1 - a^ ; x - ij hy x\ - iji. 4. (a — 6 - c)" - (a - 6 - c)"- "• - (rt — 6 — c)"" by (a- h-cyr 5. 2 2^3 + 2 y3 + 2z^- 6zyzhy (x - yf -\- (y - zf + (z-2:)2; (2^-ff by {x' + xy + ff. 6. {x^-2yzf-^f7? by ^^2-4^2; (a:+2y)8 + (y-3«)3 by ar + 3 (y - z). 78 ELEMENTS OF ALGEBRA. 7. x^ — x^i/ -\- xy^ — 2 xi y^ + y^ by xi — xy^ -\- x^y -yl 8. 2 :r3« _ 6 ,jc2nyn _j_ e ^n^2n _ 2 fn ^^ ^n _ ^n^ by :r"' - 3 a;'""^ / - 6 a:'"-^ ?/2«. 10. a3" - 3 a2« 5« + 3 a« &2« _ 53^ by ^« _ j« . 3 ^-o: - 8 a^ + 5 ^3^ - 3 a-3^ by 5 a^ - 3 ^-^ 11. 6 «'« + ' - 23 a^" + ^ + 18 a^ - a^"-^ - 3 a'"~' 4. 4a3«-3 - a""-' by 2 a^''^^ - 5 a^" - 2a2'*-^ + a''-'. 12. 4 at - 8 ai - 5 + 10 ^-^ + 3 a"! by 2 ai2 - aiV - 3 h~l 13. 6:r^+3-5:2r" + '-6 2r^ + ^+19af-21r^-^ +4,r^-2 by 2 x^ + x^ — 4:X. 14. 6m*-"-*-2 + m*-"+^ - 22 m^"" + 19 7?t*-"-^ 15. 6aj^ + "+'+2f^-"-^^-9af + "+lla;^ + "-'-6;zf + "-2 ^^ + n-3 by 2 2^'^-^'+ 3a;" + ^-a;". 16. a""* — ct" fe(''-i)'» — a<"*-i>" &"* + 5"*" by a" - &**. Find an exact divisor and the quotient of the following, by inspection : 17. 8a^+l; 16 - 81 a^ ; 64 a^ - 8 &3 . ^34. iqOO ; ^6 - 64 ; m^ -71^; 1 - 8 ?/3 ; a^ 6^ - 1. 19. 8 ;x^6 - 27 ?/-^ 64 ai2 _ 27 ^-9; 243 a^ + 32; cSjc^'* - ftS^Sm . 1^ ^4n _ 0016 hh'^', 29" al2» + 36n EVOLUTION. 79 CHAPTER VIL EVOLUTION. 37. Evolution is the operation of finding one of the equal factors of a number or expression. Evolution is the inverse of involution. By Art. 27, (2 a)» = 4 a^; (2 «)» = 8 a* ; (2 a)* =16 a*; etc. 2 a is called the secoml or sqiuire root of 4 o^ l)ecause it is one of the two ec[ual factors ot 4 a'^; it is the thii-d or cube root of 8 a* because it is one of the three equal factors of 8a^; etc. Hence, in general, A Root is one of the equal factors of the number or expression. Roots are indicated by means of fractional exponents, the denominators of which show the root to be taken. Thus, (a)* means the second or square root of a ; (a)* means the third or cube root of a; (a*)* means the sixth root of a^. In general, (a*")* means the nth root of a"*. Roots are also indicated by means of the root sign, or radical sign, ^. Thus, \/a means the square root of a ; ^a means the cube root of a ; v^ means the nth root of a"». The Index is the number written ia the opening of the radical sign to show wliat root is sought, and corresponds to the denominator of the fractional exponent. When no index is written, the square root la understood. 80 ELEMENTS OF ALGEBRA. « — 1 Note, ya or a» is defined, when n is a positive integer, as one of the n equal factors of a ; so that if Va be taken n times as a factor, the resulting product is a ; that is, ( ya)'^ or i^a" j" = a. ,mn -\mn ( -^\mn, Similarly, ( \/a) or Va*""/ = a, 38. The sign, ± or =p, is sometimes used and is called the double sign; it indicates that we may take either the sign + or the sign — . Thus, a ± 6 is read a plus or minus b. By Art. 27, (+a)4 = a^ (-a)4 = a4; (+a)5 = a5; (-ay = -aK Therefore, (a*)^ = ±a; (+ a^)^ = a; (— a^)^ = — a. Hence, in general, Hven roots of any nu7nber are either positive or negative. Odd roots of a nwniber kave the same sign as the number itself. Since no even power of a number can be negative, it follows that, An even root of a negative number is impossible. Such roots can only be indicated, and are called imaginary. Thus, (— a^)^y Y^— 6, Y^— 1, and ^— a^, are imaginary. Example 1. Find the square root of 9 a^b^c^. Solution. Since, to square a monomial, we multiply the expo- .nent of each factor by 2, to extract the square root we must divide the exponent of each factor by 2. The two equal factors of 9 are 3 X 3, or 32. Dividing the exponent of each factor by 2, we have 3 a^ b'^ c. Since the even root of a positive number is either positive or negative, the sign of the root is either plus or minus. .-. y/9aH^ = ±3a^b^c. Example 2. Find the fifth root of - 32 a^^ x"*. Solution. Since, to raise a monomial to the fifth power, we mul- tiply the exponent of each factor by 5, to extract the fifth root we must divide the exponent of each factor by 5. The equal factors of 32 are 2 X 2 X 2 X 2 X 2, or 2^. Dividing the exponent of each EVOLUTION. 81 factor by 6, we have 2 a* x*. Since the odd roots of a number have the same sign a.s the number itself, the sign of the rout is minus. .♦. J^— 32 a*® x*» = — 2 a* a*. Hence, in general, To find any Boot of a Monomial Resolve ike numerical coejffkient into its prime factors, each factor being written with its highest exponent, divide the exponent of each factor by the index of the required root, and take the ^product of the resulting factors. Give to every even root of a positive expression tlie sign ± , and to every odd root of any expres- sion the sign of the expression itself Hote. Any root of a fraction is found by taking the required root of each V27 of ito terms. Thus, .Vl _ J^ = 2 in general, t/- = -^ Exercise 31. Find the value of the following expressions : 1. V25^; ^-%aH^a^', y/-12baH^', v^81ai«W 2. (-343a*5-6)i; {\{7^ifz^)^; {-x^^y^^)^- v'Sl^V^. 3. v^iw. (I21a:i2y2)i. V25 aH"^-, (16a-8&8)i 4. (- 243 a^* Jio«)i. (_ 54 ,,,3 ^6 ^^J. (^w ^80)tV. 5. (-32ai0y-5)i; V^2W^r^ -, (625 a^ 6i« c*)i. 6. (512an8ci5rf-8)i; V64a-«&"*; -y/m ; ^/- 32 ai^. 8. V'i^^; (2"a2«54n2^)i; V8l2:5i-y2m+4. >^_ 8 a;8*-«ye-+». A 82 * ELEMENTS OF ALGEBRA. 10. \/l6x^''y^''z^; (- iV^ ^^- ^ ?i" ^M ; V^-^^Sm^-m Simplify : 11. Sj^a^hU-l + iij a^ hh d)^ - (f i a* ol c-^)i — V ^^ <^^« c 4. V 50 Express the nth roots of: 12. 3x7x4; 52:"2/2^ 3a^&3. (a-ir)3; (^^z)"; ^'"-y"; Express by means of exponents : 13. \/JWc^; 7a(x-yT; V^^^; V^(^ + 2/)~ Queries. If n and j9 in the last two parts of Ex. 13 are integral, what signs should the roots have ? Why ? When should the first two roots have the double sign ? 39. By Art. 28, (a + hy = a^ + 2ah -\- b^. Therefore, (a^ + 2 a 6 + 62)i = a-^b. By observing the manner in which a + b may be obtained from a^ + 2ab + b^, we shall be led to a general method for finding the square root of any polynomial. Process. a"^ + 2ab -h b^(a + b First term of the root squared, a^ First remainder, 2 ab + b^ Trial divisor, 2 a Complete divisor, 2a + b Complete divisor X b, 2ab + b^ Explanation. The square root of the first term is a, which is the first term of the required root. Subtracting its square from the given expression, the remainder is 2ab + b^, or b times 2 a -\- b. EVOLUTION. 83 Since the first terra of the i-einainder is twice the product of the fii-st and last terras of the root, and we have found the first terra ; there- fore, divide 2 a6 by twice the fii-st terra of the root already found, or 2 a. 'I'he result will be the second terra b of the required root. Adding 6 to the trial divisor gives the complete divisor, 2 a + 6. Multiplying by 6 and subtracting, there is no remainder. By Art 28, (a-\-b+cy = a^+2ab-{-b^+2ac + 2bc + c^. Therefore, (a^-{-2ab-^b^-\-2ac-^2bc + c^^ = a + b+c. Process. a^-\-2ab-\-b^+2ac-^2bc + c'^{a-\-b + c First terra of root squared, a^ First remainder, 2ab + b^-{-2ac + 2bc+c^ First trial divisor, 2 a First complete divisor, 2a-\-b First complete divisor X 6, 2ab-{-b^ Second remainder, 2ac-\-2bc-^c* Second trial divisor, 2a-|-26 I Second complete divisor, 2a-\-2b-i-c \ Second complete divisor X c, 2ac+2bc-\-c^ Xtxplanation. Proceeding as before, the first two terms of the root are found to be a + 6. To find the last terra of the root, take twice the terms of the root already found for the second trial divisor. Dividing 2 a c by the first terra, the result c will be the third term of the required root. Adding this to the trial divisor, gives the entire divi.<*or. Multiplying by c and subtracting there is no remainder. We have actually squared the root and subtracted the square from the given expression. Hence, in general, To find the Square Root of any Polynomial Arrange the terms according to the powers of one letter. Find the square root of the first term. This will be the first term of the required root. Subtract its sciuare from the given expression. Divide the first term of the remainder by twice the root already found. The quotient will be the next term of the root. Add the quotient to the divisor. Multiply the complete divisor by this terra of the root, and subtract the product frora the remainder. For the next trial divisor, take two times the terms of the root already found. Continue in this manner until there i-* no remainder. 84 ELEMENTS OF ALGEBRA. Example. Find the square root of 4 a — 10 a^ + a^ + 4 a^ + 1. Arranging according to the ascending powers of a, we have, [_2a2-a3 Process. First term of root squared, First remainder, First trial divisor, 2 First complete divisoi' , 2+2a 2 a times first complete divisor, Second remainder, Second trial divisor, 2+4 a Second complete divisor, 2+4 a— 2 a^ 1+4 a -10a3+4a5+a6(i4-2a _1 4a-10a3+4a5+a6 4a^+4q -4a'-^-10a3+4a5+a« — 2a^ times second complete divisor, Third remainder, Third trial divisor, 2+4 a— 4 a^ Third complete divisor, 2+4 a— 4 a^—cfi -4a2-8a8+4a^ -2a8-4a4+4a6-(-a« a* times third complete divisor. ~2a3-4a4+4a5+a« Note. The student should notice that the sum of the several subtrahends is the square of the root, and that he has actually squared the root and sub- tracted the square from the given expression. Exercise 32. Find the square roots of: 1. 2^-4^3+ (]y2_4y4.i. 9^4- 12 a3_ 2^2+4^ + 1. 2. 4a6_ l2a^l-~ 11 a^^ly^ + b^ a^ h^ - 17 aH^- 70 ab^ + 49 &6. 3. x^ - 12 a;5 + 60 x^ - 160 a^ + 240 x'^ - 192 a; + 64. 4. 8 a + 4 + a^ + 4 ^3 +- 8 a2 5. 9 +- a;6 4- 30 :r - 4 a^ + 13 x^ + Ux^ - Ux^, 6. 6a62c_4a2&c + ^252 + 4ft2c2 + 9 52^2 - 12 a.&c2 7. 49 a^ - 28 a^ - 17 a^ + 6 « 4- f. EVOLUTION. 85 8. 4x^ + 9 y^ + 25 cc^^ 12x1/ -SO ay- 20 ax, 9. VL^ — 6 a 771^ + 15 a2 7n* — 20 a^ vi^ + 15 a* m^ — 6a^7n 10. 1 -2 a + 3 a2 - 4 a3 4- 5 rt* - 4 a5+ 3 a«— 2 a^ + a^. 11. 9 vi^ — G 7« n + 30mx+6my-\-7i^—i0nx—2 ny + 25r»+ lUa;^+ //2. 12. 7^-\-l5x^{/^^- 15aj*//2 4. ,/6_ ^y _ 20xV- <^^"^y- 13. 49x^/- 24x^/3 -30 ar8y + 25 a^+ 163^. 14. a;6 - G 2:^ + 172:4 - 34 2:3 _^ 40 a:2 _ 40 2: + 35. 15. 4 - IG «t + IG a^ + 12 rf - 24 at ?>i + 9 h. 10. ^a:4_|^y^_^,,2,,2_,^y^^,^4. 2:4«_ 0a;3n+ 5^* 4- 12 a;" + 4. 17. 25 ^1 + 1 G - 30 X -24:xh + 49 .rf 18. 9x-2+ 122;-V^-6a; + 42^-4ar^7/^ + ^. 40. Since the square root of an expression is either + or — , the square root o[ a^ -\- '2 a h -\- h^ is either a + h or — a — h. In the process of finding the sfjuare root of a'* + 2 a 6 + />', we herein by tak- ing the square root of a*, and this is either + a or — a. If we take — a, and continue the work as in Art. 39, we get for the root -^a-^h. Also, the square root of a' — 2 a 6 4- />^ is either a — h ov — a -{- b. This is true for every even root. Hence, the signs of all the terms oj an even root may he changed^ and the number will still be the root oj the same expression. Thus, last process Art. 39, if — 1 he taken for the square root of 1 we shall arrive at the result — 1 — 2 a + 2 a'* 4- a*. 41. Square Root of Numerical Numbers. The method for extracting the square root of arithmetical numbers is based upon the algebraic method. 86 ELEMENTS OF ALGEBRA. Since the square root of 100 is 10, of 10000 is 100, etc., it fol- lows that the integral part of the square root of numbers less than 100 has one figure, of numbers between 100 and 10000 two figures, and so on. Hence, If a point he placed over every second figure in any numher, begin- ning with units' place, the numher oj points ivill show the numher of figures in the square root. Thus, the square root of 324947 has three figures ; the square root of 441 has two figures. If the given number contains decimals, the number of decimal places in the square root will be one half as many as in the given number itself. Thus, if 2.39 be the square root, the number will be 5.Vi2i; if .239 be the root, the number will be 6.057121 ; if 10.321 be the root, the number will be 106.523041. Hence, The numher of points to the left of the decimal point will shorn the numher of integral places in the root, and the numher of points to the right will show the numher of decimal places. Example 1. Find the square root of 45796. a +&+c = 214 Process. 45796(200+10+4 = 214 The square of a or 200, - 40000 First remainder, 5796 First trial divisor, 2 a, or 400 I First complete divisor, 2a-lh, or 410 [ First complete divisor X &, or 10, 4100 Second remainder, 1696 Second trial divisor, 2a+2&, or 420 i Second complete divisor , 2a+2&+c, or 424 | Second complete divisor X c, or 4, 1696 Explanation. There will be three figures in the root. Let a -^h -\- c denote the root, a being the value of the number in the hundreds' place, h of that in the tens' place, and c the number in the units' place. Then a must be the greatest multiple of 100 whose square is less than 45796, this is 200. Subtract a^, or the square of 200 from the given number. Dividing the first remainder by 2 a, or 400, gives 10 EVOLUTION. 87 for the value of b. Add this to 400, multiply the result by 10 and subtract. Dividing the second remainder by 2 a -f 2 6, or 420, gives 4 for the value of c. Adding this to 420, multiplying and subtract- ing, there is no remainder. Hence, 214 is the required root; because we have actually squared it and subtracted this square from the given number and found no remainder. The student should observe that the .sum of the several subtrahends is the square of the root. Example 2. Find the square root of 17.3 to lour decimal places. Process. S<iuare of 4, First remainder, First trial divisor, 8 First complete divisor, 81 First complete divisor multiplied by 1, Second remainder. Second trial divisor, 82 Second complete divisor, 825 17.360<KK)06(4.1693..., 16 130 81 4900 Second complete divisor nmltiplied by 5, Third remainder, Third trial divisor, 830 Third complete divisor, 8309 Third complete divisor multiplied by 9, Fourth remainder. Fourth trial divisor, 8318 I Fourth complete divisor, 83183 | Fourth complete divisor multiplied by 3, Fifth remainder, 4125 77500 74781 271900 249549 22351 Let the student formulate a method for arithmetical square root from what has been demonstrated. NotM ; 1. If the trial divisor is not contained in the remainder, annex to the root, also to the divisor, then annex the next period and divide. 2. Should it be found that after completing the trial divisor, it gives a pro- duct greater than the remainder, the quotient is too large, and a less quotient must be taken. 3. Tf the last remainder is not a perfect square, annex periods of ciphers and procee<l as before. 4. The square root of a fraction may be found by taking the square root of its terms, or by first reducing it to a decimal. 88 ELEMENTS OF ALGEBRA. Exercise 33. Find the square roots of : 1. 33124; 41.2164; Jf | ; ^%^^^; .099225; 1.170724. 2. .30858025; 5687573056; 943042681. Find the square root to four decimal places of: 3. .081; .9; .001; .144; if; .00028561; 3.25; 20.911. 42. By Art. 29, (a + by = a^ + 3 a^ b + 3 ab^ + bK Therefore, (a^ + 3 a^ 6 + 3 a &2 + b^)l = a + b. By observing the manner in which a -\-b may be obtained from a^ + 3 a^ft + 3 a &2 -I- 6"^, we shall be led to a general method for find- ing the cube root of any compound expression. Process. a^+2a%yMb'^-\-b^ {a+b First term of the root cubed, o^ First remainder, 3a%+Zab'^-{-b^ Trial divisor, or 3 times the square of a, Za^ 3 times the product of a and b, 2ab Second term of the root squared, 6^ Complete divisor, Za^+Zab+b'^ Complete divisor X &, Za%+Zab'^^-b^ Explanation. The cube root of the first term is a, which is the first term of the required root. Subtracting its cube from the given expression, the remainder hZaH + Zab'^+b'^, or b times Za^ + Zab+b^ Since the first term of the remainder is three times the product of the square of the first term of the root multiplied by the last term, divide Za^b by three times the square of the first term of the root already found. The result will be the second term 6 of the required root. Adding to the trial divisor three times the product of the first and second terms of the root, and the square of the second term, gives the complete divisor, ov Zw^^ Zab + b^. Multiplying by b and subtract- ing, there is no remainder. Since the cube of a+b-\-c is a^-\-Za'^b^-Zab'^-\-h^+Za^c + Qabc -f-3&2c + 3ac2+36cHc8, the cube root of a8 + 30^6 -|- 3 a 62+63+ 3 a^c + 6a6c + 362c+3acH36c2+c« is a + b + c. EVOLUTION. 89 i CO 4- ^ ^j o eo cS j; w ^ 1 t s 5 -- ^ ^ ^ t ^ 1 •o e^ « $ # c o J cS <Sc^'" ^5 ^ o I « ^ X ^ iz •5 S ,2 O C ;:2 'tis -k.) I- Ph C ■2 5 -^ -r -^ -« -^ •> g ^ 3 B .5 -^ "^ q O) 4> eS ^_^ •2 iJ s S S is Ts £ -3 8 8 -2 '^ § £ I ^ ^ § §^ 3,3. s a 8 8 o p 90 ELEMENTS OF ALGEBRA. Explanation. Proceeding as before, the first two terms of the root are found to be a + h. To find the last term of the root, take three times the square of the terms of the root already found for the second trial divisor, and divide Za^c by the first term. The result will be the third term of the required root. Adding to the second trial divisor three times the product of a + & and c, and the square of c, gives the second complete divisor. Multiplying by c and subtract- ing, there is no remainder. Observe that the sum of the several sub- trahends is the cube of the root, and that we have actually cubed the root and subtracted the cube from the given expression. Hence, in general, To find the Cube Root of any Polynomial. Arrange the terms according to the powers of one letter. Find the cube root of the first term. This will be the first term of the required root. Subtract its cube from the given expression. Divide the first term of the remainder by three times the square of the root already found. The quotient will be the next term of the root. Add to the trial divisor three times the product of the first and second terms of the root, and the square of the second term. Multiply the complete divisor by this term of the root, and subtract the product from the remainder. For the next trial divisor, take three times the square of the root already found. Continue in this manner until there is no remainder or an approximate root found. A Term may be a figure, or a letter, or a combination of figures and letters, or of letters only, produced by multi- plication or division, or both. Thus, in the algebraic expression 5 + 2a®6* — a+ -gi^; 5, 2a^b*, a, ah- "^y «2 ,,w are terms. An Algebraic Expression is a representation of a number by any combination of algebraic symbols. Example. Find the cube root of 27 a — 8 a^- 36+ 36 a^- 12a- ^ -54a^ + 9a-§+27a§ + a-6-6a-J. The work is conveniently arranged as follows : EVOLUTION. 91 + I 1 + + •4M •* HB« o l„ 1 <N C3 o J CO «o CO 1 1 1 1 **— ^ ^ e C5 05 4- + r* k 1 I 1— • 1 CO 1 1 o 1 t^ r^ 04 (N + + ■an im c2 1 <2 1 1 1 5 ^ CO CO + + »«• HM 1 1 1 1 1 C 8 t- I- <M CN + Mi CD I I 4- 4, ♦- j-J 4, O) 02 CO H cc cw 92 ELEMENTS OF ALGEBRA. Exercise 34. Find the cube roots of : 1. x^— Sx^ + 5x^- 3x --1; x^- Saa^+ 5 a^ a^ — S a^ X — a^. 2. 8 0^6 + 48 ax^+60 a^x'^-80 a^x^- 90 a^x^+lOSa^x - 27 a^. 3. x^-6x^+ 15 x^-20 a^+15x^-6x+l. 4. 27a^-54.a^h + 9a^^+2SaH^-3a^b^-6ah^-h^ 5. Sx^+ 12 x^ -SOx*- 35 x^ + 4:5x^ + 27 x - 27, 6. 216 + 3422:2+i7i^4 4.27^6_27^5_i09^-108ic. 7. a3 - 3^25 - 53+ 8c3+ 6a2c-12a&c + 6fe2c 4-12ac2- 125c2+ 3ah\ 8. 1 - 3rK + 62^2 - 10 rt3 + 12 ic^ - 12 a;^ + 10 ri;^ - 6 a;^ 4- 3 a:-8 - .^'9. 9. 8 x^ - 36 x^y + 114 aj*?/^ _ 207 x^y^ + 285 0^23^ -225 0^2/5+125 7/6. 10. a^ + 6a^h - Sa^c + 12 a V^ - 12 abc + 3ac2 + 8^3- 12h^c+ 6bc^-c^. 11. x^ + Sx^y-Sa^y^-~lla^f+6xh/^+12xf - 8/. 12. 204 rr*2/2 - 144 a;^ 7/ + 8 /- 36 o;^/^ _ 171 ^^s^^s _|. 54^6 + 102 0^2/*. 43. Cube Root of Numerical Numbers. The method for extracting the cube root of arithmetical numbers is based upon the algebraic method. EVOLUTION. 93 Since the cube root of 1000 is 10 ; of 1000000 is 100, etc., it fol- lows that the integral part of the cube root of numbers less than KXX) has one figure, of nuiubers between 1000 and 1000000 two figures, and so on. Hence, I/a point be placed over every third fgure in any number, beginning with units' place y the number of points will show the number of figures in the cube root. Thus, the cube root of 274625 has two figures ; the cube root of 109215352 has three figures. If the given number contains decimals, the number of decimal places in the cube root will be one third as many as in the given number itself. Thus, if 1,11 be the cube root, the number will be 1.367631 ; if .111 be the root, the number will be (3.00i36763i ; if 11.111 be the root, the number will be 1371.706960631. Hence, The number of points to the left of the decimal point will show the number of integral places in the root, and the number of points to the right will show the number of decimal places. Example 1 . Find the cube root of 778688. a + 6 = 92 Process. 778688 ( 90 + 2 -= 92 The cube of a, or 90, 729000 First remainder, 49688 First trial divisor 3 a", or 3 (90)^ = 24300 3 times the protluctof a and 6, or 3X90X2= 540 Second term b of the root squared, 2*'* = First complete divisor, 24844 First complete divisor X 6, or 2, 49688 Explanation. There will be two ^gures in the root. Let a -f- 6 denote the root, a being the value of the number in tens* place, and b the number in units' place. Then a must be the greatest nmltiple of 10 whose cube is less than 778688, this is 90.. Subtract a«, or the cu>)e of 90, from the given number. Dividing the remainder by 3 a*, or 24.3()0, gives 2 for the value of 6. Add to the trial divisor 3 a 6, or 54(), and 6^, or 4, for the complete divisor. Multiplying by 2 and subtracting, there is no remainder. Hence, 92 is the required 94 ELEMENTS OF ALGEBRA. root, because we have actually cubed it and subtracted this cube from the given number and found no remainder. Example 2. Find the cube root of 897.236011125. = 24300 1620 36 25956 Process. Cube of 9, First remainder, First trial divisor, 3 times (90) ^ 3 times the product of 90 and 6, 6 squared, First complete divisor, First complete divisor multiplied by 6, Second remainder. Second trial divisor, 3 times (960)2 - 2764800 3 times the product of 960 and 4, 11520 4 squared, 16 Second complete divisor, 2776336 Second complete divisor multiplied by 4, Third remainder, Third trial divisor, 3 times (9640)2 = 278788800 3 times the product of 9640 and 5, 144600 6 squared, 25 Third complete divisor, 278933425 Third complete divisor multiplied by 5, 897.236011125(9.645 729 168236 155736 12500011 11105344 1394667125 1394667125 Let the student formulate a method for arithmetical cube root from what has been demonstrated. Note. The notes in Art. 41 are equally applicable to cube root, except that in Note 1 two ciphers must be annexed to the divisor instead of one. Exercise 35. Find the cube roots of: 1. 74088; 34012.224; .244140626. 2. ^mif^; .000152273304. EVOLUTION. 95 Find to three places of decimals the cube roots of : 3. .64; .08; 8.21; .3; .008; J; ^. 44. Since a* = c^^ = (a*)* = Vo* = VVa, The fourth root is the square root of the square root. * Since a* = a^ '<3 = (aSf = Va* = V 4^, The sixth root is the cube root of the square root. Hence, When the root indices are composed of factors, the ope- ration is performed by successive extraction of simpler roots. Hote. It is suggested that the teacher use the remainder of this article at his discretion. We may find the fifth, seventh, eleventh, or any root of an expression or arithmetical number if desired, by using the form for completing the divisor. Thus, To find the fifth root. Form, (a + 6)» = a* + (5 a* + 10 a«6 + 10 a*^** -j- 5 a 6» + 6*) b. Trial divisor, 5 a*. Complete divisor, (5 a* -h 10 a» 6 + 10 a* &« + 5 a 6» + b*) . To find the seventh root. Form, (a+6)' = a7+(7a»+21a»H35a<6«-f35a»6»+21a26<+7a6»+6«)6. Trial divisor, 7 a*. Complete divisor, {7a*+2la^b+35a*b^+3ba%^-2la^b^-7ab^+h% 96 ELEMENTS OF ALGEBRA. Example. Find the fifth root of 36936242722357. Process. 36936242722357(517 a5 = 5S = 3125 First remainder, 56862427 First trial divisor =:5a'^(a considered as 5 tens) = 5 (50)* = 31250000 10 a^b (h considered as 1 unit) = 10(50)3 X 1 ::= 1250000 10a2 62= 10 X (50)-^X (1)^ = 25000 5 a 63 = 5X (50) X (l)^ ^= 250 6* = (1)^ 1 First complete divisor, 32525251 First complete divisor multiplied by 1, 32525251 Second remainder. 2433717622357 Second trial divisor = 5 a^ (a considered as 51 tens) = 5 X (510)^ = 338260050000 10 a^b{b considered as 7 units) = 10 X (510)3 X (7) = lOa-^62 = 10 X (510)2 X (7)2 = 5a68 = 5X(510) X (7)3 = 6* = (7)4 Second complete divisor, 347673946051 Second complete divisor multiplied by 7, 2433717622357 9285570000 127449000 874650 2401 Miscellaneous Exercise 36. Express the nth. roots of: 1. ah^c-'^; 52-a;3"(^-;y + 2«)4"x2%^4-7/"y"x4"(a;— 2/T- 2. Simplify 4: a (Sax y)^ — 5 xhj^ ^2S^ «^ X ak Find the square roots of: 3. ^x + l^-\-x-^—4x-^^-l2x^. 4. 28-24tt-^-16al + 9ft-" + 4a^ EVOLUTION. 97 5. 162;'^"+16ic7--4a;8*-4x-9'' + a:io». 6. a^u~l—4:xhj~^ + 6 — 4 x~^ yi + x~^t/^. 7. 6«cr/;5 + 4 62u;* + a^x^^ -^ \) c^ -12 bcx^ - 4: abx'. 8. {x'^ + Ax^ + ^ax^ -^ ia^-2x^-^ ax. 9. a2»+ 2a\6"'+ z'""; « ± 2 ai a;i + x. Find the cube roots of : 10. 60 x^i/ -^ 4S xf ^27 x^ + lOS x^i/ -90 si^y^ -{- 8/ -80a^^. 11. 24a;*'" 2/2- 4- % 3^'^yin_ 6a^'»7/" + a:6m_9g^^n + 64/"- 56a;3«2^* 12. 15x-* - 6:r-i— 62;-6 + 15aj-2+ 1 + a:-6-^20a;-8. 13. Su^-^2^f + ixi/-^f. 14. ^a-i-6a-J-J + 8a-t-^a-8 + 27a-i + 54a-4 + |a-t + 36 a-4- 18a-2. Find the sixth roots of: 15. 1215a*-1458a6+135a2-540a3_l8« + l+729a« 16. .x^+ f-6x ?/+ 15jy^ij^-Gx^y+ 15 3^1/- 20 a^y^l 17. 160 a3 + 240 a* + 60 a2 + 192 a^ + 64 a«+ 12 a + 1. 18. 2985984; 262144. Find the eighth roots of : 19. </8+28a3+8a+l + 56a8+70a* + 8aH56«6^28a«. 20. («* + 5* - 2 a&8 + 3 a2i^ - 2 aH)*. 7 98 ELEMENTS OF ALGEBRA. -^ 21. Find the 5tli root of 36936242722357. 22. Find the 7th root of 1231171548132409344. Extract the following roots : 23. y/(a4 + 19 a2 + 25 - 6 aS _ 30 a), 24. y/[^^ — 2 {m + 71)0^ + (m^ + 4:mn -\- n^) x^ — 2mn (m -\- n) x + 7/^^ 71^] 2 25. [25 a2 _ 20 a 5 + 4 52 + 9 c2 - 12 6c + 30 ac]i 26. [27a^- 54a5+ 63 a^-Ua^^ 21a2_6a + l]i 27. y/(aj2'" + 2 af"-^'^ - 2 ^'"^^ + ^2» _ 2 aj"+^ + ^). 28. [a6_ 12^5+ 60^4- 160^3+ 240 a^-192a+ 64:]^. 29. [(«4-fe)6'»^34.6^n^(^^j^4m^_^^2rt2V(a+&)2'"ic+8a3V]i 30. y/(^2« ^ 2 a:2«-l + 3 3^2—2 _^ 2 a;2'»-3 + aj2n-4)^ 31. y^(8 - 12a3«-i + 6a6"-2_a9«-3^^ Queries. What signs are given to even and odd roots ? Why ? What principles govern the signs of roots ? Upon what principle is the method lor finding the root of a monomial hased 1 How derive the method for finding the square root of any polynomial '? Why divide the first term of the remainder hy twice the terms of the root already found for the next term of the root ? Why add the quo- tient to the trial divisor for the complete divisor ? How derive the method for finding the cuhe root of any polynomial ? Why divide the first term of the remainder by three times the square of the root already found for the next term of the root ? Why add to the trial divisor three times the product of the terms of the root already found by the next term, and the square of the next term, for the complete divisor ? USE OF ALGEBRAIC S\MBULS. 99 CHAPTER VIII. USE OF ALGEBRAIC SYMBOLS. 45. Symbols of operation are used to indicate that algebraic operations are to be performed. Thus, m -f (a - 6) indicates that a — 6 is to be added to m ; m — (a — 6) indicates that a — h is to be subtracted from m. Per- forming the operations, we have, m-\- (a — b) = m + a — b ; m — (a — h) = m — a + b. Hence, A plus sign before a symbol of aggregation shows that the enclosed terms are to be added to what precedes ; as this operation does not change the signs, the removal of the symbol does not affect the signs. Removing one preceded by a minus sign changes the sign of each enclosed term. Thus, a-26-[4a-66-{3a-c+(5a-26-3a-c + 2 6){] = a-26-[4a-66-{3a-c + (5a-26-3a + c-2 6)}] = a-2 6-[4a-66-{3a-c+(2a-46 +c )}] = a-26-[4a-66-{3a-c+ 2a-46 +c |] = a-26-[4a-66-{6a -46 }] = a-26-[4a-66-5a +46 ] =a-26-[ -26- o ] = a-26 +26+ a = 2a Explanation. Remove the vinculum, subtract and unite like terras ; then remove the parenthesis and unite like terms ; now remove the brace, subtract and unite like terms ; finally, removing the bracket, subtracting and uniting like terms, we have 2 a. 100 ELEMENTS OF ALGEBRA. Exercise 37. Simplify : 1. 2a-l3b+(2b~c)-4:c+ {2 a-(Sh-c-2h)}']. 2. a — b + c - (a + b — c) — (c — b- a). 3. x^ ~[4x^~ {6 x^ - (4.r - 1)}] -(x^ + 4:x^+6x^ H- 4^ + 1). 4. ~l0(x-\-y)-lz + x + y-3{x + 2y-(z+x-y)}^ 4- 4:Z. 5. a-[5b-{a-{5c-2c-b-4b) + 2a-{a-2b-hc)}l 6. -5{a-6[a-{b- c)]} + 60 {6 - (c + a)}. 7.-2a-(3b + 2c)-[5b-{6c-U) + 5c-{2a-(c+2b)}]. 8. 3:r - {?/ - [?/ - (:r + ?y) - {- ;/ - (^^ _ ^ - ^z)}]}. 9. 3a-[2b + a-b]-^ [Sb- 2a + bl 10. {(re- 2 ?/ + a; ?/) - (a; - ?/ + ^)}-{a: -(x-2j + xy)}. 11. I a - [| a - {1 « - (2 ft - 5 a + 6)} - (f a - 3)]. 12. f{f(a-6)-8(&-c)}-{|(6-0-i(c-«)} - I {^ - a - |(« - 6)}. 13. 5 {a - 2 [ft - 2 (ft + a;)]} - 4 {« - 2[a- 2 (a + rr)]}. 14. ft+25-{6a~[3& + (8a;-2 + &.y-a; + 4a)]-3/;} + 2(1 + |-«-46). 15. 2 (|& - f ft) ~ 7 [ft - 6 {2 - 5 (ft - b}}\ 16. -|{-f[-4(-:.i)]}+f{-3(-a;^)}. 17. - I {- L" Q^ - h)]} + {- I [- (ft - fe)]}. USE OF ALGEBRAIC SYMBOLS. 101 18. 5 {a - 2[6 - 3 (c H- d)]} - 4 {a - 3 [6 - 4 (c - </)]}. 19. (a-l)(a - 2) - 3a(a + 3) + 2 {(a + 2)(a + 1)- 3}. 20. {xz -{x'-y){y + z)} -y[y-{x- z)\ 21. {a^h-\-c-\-df^{a-h-c^ df+ (a-b + c-d)^ + (a + b-c- d)\ 22. 7!'-{2xy-[-{x-{y-z}){x+{y-z})-\-2xy-\-4.yz} 23. a(a + l)(aH-2)(a + 3)-6(2a-J)-"ta2-3a + l)2. 24. 571 {(a:— y)a-&2;}— 27i{a:(a-6)-af ?/} - {Zax—{pzr-2x)h} n. 25. (a:2 4.2^)^_(a; + y)(a;{7i-y}-y{^i-4). 26. 2aVi — 3 m — [6yi — 6 7i + (2:i-2Vi)a] + &X Vy. 27. (9 7?l2 7l2 - 4 71*) (7m2 - 7l2) - {3 77? 7?. - 2 W^} {3 771 (/?t2 4- 71^) — 2 71 (71* + 3 771 71 — TH?)} 71. 28. 77l2 ( w2 + 7l2)2 - 2 7^2 ^2 (^ + n) (771 - 7^) - (tTI^ _ n3)2.. 29. i(^^ + iy)(i^-Jy)-(J^-!.y)^-?(^-f2/^). 30. Y a ^ + 4 ?/') ( J ^ - J ?/') - (i :r - 3) (J :r + 3) (4^-9) + (f y - 3)(f y + 3) (^ .7/2 - 9). The use of symbols of aggregation aid in shortening the work in certain cases in division. Thus, a 4- (6 + c) ) (6 + c) a^ + (62 -I- 6 c + c') a - - (6 + c) 6 c ( (6 + c) a - 6c (6 + c)a« + (62+ 2 6c + c«) a + ( - 6c )a-(6 + c)6 + ( - 6c )a-(6 + c)6 102 ELEMENTS OF ALGEBRA. Divide : 31. (6 + c)a2 + (&2 + 32^c + c2)a+(i + c)&c by a + h + c. 32. (a -\-hf-6{a + b)- 27 by (a + b) + 3. 33. {x+ijf+ 3 {x + yfz + S{x + y)z^+^ by (x + yf + 2(x + y)z + z^. 34. (x + yf _. 2 (^ + ?/) z + z^ by x ^ y — z. 35. {a + &)3 + 1 by « + J + 1. 46. The converse operation of enclosing any number of terms of an expression in a symbol of aggregation is important. a + m~c-\-h — n = a + m — c-\-(h — 71). a — m — c -]- b ~ n = a — (m + c) -^ (b — n). ax^ — ny + hx^ — cy^= {ax^ — ny) + (hx^ — c y*). xy — ax'-by + ab = (x y — by) — (a x — a b). Hence, when the signs + and — indicate operation : (1) Any number of terms may be enclosed in a symbol of aggrega- tion preceded by the sign + , without changing the sign of each term. (2) Any number of terms may be enclosed in a symbol of aggrega- tion preceded by the sign — , if the sign of each term be changed. The terms may "be enclosed in various ways. Thus, am + an — ax — bx + cy — dz=(am — ax) + \an — bx^-\-\cy — dz\, or, am-\-an—ax — bx-\-cy—dz= (am, + an — ax)—{bx — cy + dz], or, am + an — ax — bx+cy — dz= {am-\-an)—{ax-\-bx)+{cy—dz). Etc. If a factor is common to each term within a symbol of aggregation, it may be placed outside as a multiplier. Thus, ax^ + bx'^~bx^ + dx'^={ax^--bx^) + {bx'^+dx^)=x^{a-b)+x'^{b + d). Note. An expression consisting of three or more terms may be raised toa given power by inspection, by first changing it to the form of a binomial. Thus, (a + & + c — (^)4 = [(a + J) + (c — d)Y^ = etc. USE OF ALGEBRAIC SYMBOLS. 103 Exercise 38. Bracket the last three terms so that each bracket shall be preceded by a — si<,'n : I. a:* - a x8 - 5 a^» + 2 ; m6 + 3 w3 + 3 - 6 m2. 3. 4:X+'dao(^—{)7^ — bey -^ y; x^—y^—z^-\-ab-\-3ac. 4. Express each of the above as binomials, and enclose the last two terms in an inner brace preceded by a — sign. Bracket the following in binomials, also in trinomials, each preceded by a — sign : 5. 2ab — Say-\-4:bz — 5bx-2cd — 3. 6. a— 26 + cz — d — l-^z — x—2y+ 2m — n-\-p—4ahc. 7. 2x-Sxy-{- 4 ^f - 5 2:3^2 + ^,p _ ^.y 2 8. a^+3 a*— 4 a^—3 CT^-f a —1 ; —2 m—2> 7i+4p— 5 2;— 1—6 y. 9. a n ■\- ab — a c — c X — a X — a y — 3 ab c -\- Z xy z. 10. Express the above six examples in trinomials, and enclose the last two terms in an inner bracket preceded by a — sign. II. Expand (tw + 2 71 — j^. 12. Simplify and bracket like powers of x in 2b^—ax -{a:i^-\bx-nx- {a^ -^ 3r.i2}-| _ {ax'-2cx)}. Queries. Why may a 8ymlx)l of aggregation preceded by a + sigu be remove<l without changing the signs of the enclosed terms ? If a symbol of aggregation pi*ecedetl by a — sign be removed, why change the signs of the enclosed terms? 104 ELEMENTS OF ALGEBRA. CHAPTEE IX. SIMPLE EQUATIONS. 47. 3 a: + 5 = 5 a; — 7 is called an Equation. The first memher or first side is 3^ + 5, and the second member or second side is 5 x — 7. x = x, 14 = 14, are called Identities or Identical Equations. To solve an equation is to find the value of the unknown number. The process of solving an equation depends upon the following axioms: 1. If to equal numhers we add equal numbers, the sums are equal. 2. If from equal numhers we subtract equal numbers, the remainders are equal. 3. If equal numbers are multiplied by equal numbers, the products are equal. 4. If equal numhers are divided by equal numbers^ the quotients are equal. Example 1 . Find the value of x in the equation 6 a:— 11 — Sx+lO. Solution. Subtracting 3 x from each member of the equation (Axiom 2), we have 6a; — 3a;— ll = 3cc — 3.X+10. Uniting like terms, 3 a; — 11 = 10. Adding 11 to each member (Axiom 1), and uniting like terms, we get 3 a; = 21. Dividing both members by 3 (Axiom 4) gives x—1. Proof. To verify this result, substitute 7 for x in the given equa- tion. Then, 6 X 7 - 11 = 3 X 7 + 10, or 31 = 31, which is an identity. Hence, the value of x is 7. SIMPLE EQUATIONS. 105 Example 2. Solve the equation 2 (a: - 8) - 3 (9 - x) + 5 (a: - 1 1) = 7 -3 (a: -17). Solution. Performing the indicated operations, and uniting like terms, 10 a; — 98 = 58 — 3 z. Adding 3 x and 98 to each member of the equation, we have 10 « + 3 a: - 98 + 98 = 58 + 98 - 3 a: + 3 x, or uniting like terms, 13 x= 156. Dividing both members by 13, X- 12. Proof. Substitute 12 for x in the given equation. Then, 2(12-«) - 3 (9 - 12^ + 5 (12 - 11) = 7 - 3 (12 - 17), or, 8 + 9 + 5 = 7 + 15, or, 22 = 22, an identity. Therefore, the value of x is 12. Example 3. Solve the equation 14 — x — 5 (x — 3) (x -f 2) + (5 - x) (4 - 5 x) = 45 X - 76. Process. Simplify, 64 - 25 x = 45 x - 76. Subtract 45 x, 64 - 70 x = - 76. Subtract 64, - 70 x = - 140. Divide by — 70, x = 2. Notet : 1. To verify, that is, to jncne the truth of the result^ substitute the supposed value of the unknown number in the given equation and thus find if it satisfies its conditions. 2. In simplifying an equation the student should be careful to notice that when the sign — precedes a term, in removing the symbol of rggregation, the sign of each term must be changed. Exercise 39. Solve the following equation.s : 1. 6a;+ 1 = 5a:-f 10; 11 - 7z = 18a:- 14. 3. 2r+3= ir,-(2ar-3); 3 (a:- 2) + 4 = 4 (3 - a:). 4. 7(a:- 18) = 3(x- 14); 7a:+6-3a: = 56 + 2.r. 5. 15 (x - 1) + 4 (x + 3) = 2{x+ 7). 6. 5 - 3 (4 - a:) + 4 (3 - 2 x) = 0. 106 ELEMENTS OF ALGEBRA. 48. If we add the same number to each member of an equation, or subtract it from each member, the results are equal, each to each. Thus, Consider the equation x — b = a. Adding b to each side, we get, X = a -{- b. Consider the equation x + b = a. Subtracting 6 from each side, we have x = a — b. In each case b is transposed from one side to the other, but its sign is changed. Hence, Any term may he transposed from one side of an equation to the other, provided its sign be changed. Example. Solve (a: + 1) (x + 2) (x + Q) - (x - 2) (x -\- 2) = x^ + 9x^ + 4(7x-l)-{- (2-x)(3 + x). Process. Simphfy, x^ + 8x^ -}-20x-\-\6 = x» + 8x^-}-27x + 2. Transpose, x^ - x^ + 8x^ ~ 8x^ + 20 x-27 x = 2 - 16. Unite Hke terms, — 7 x = — 14. Divide by — 7, x = 2. * Hence, in general, To Solve a Simple Equation of one Unknown Number. If necessary, simplify the equation. Transpose all the terms containing the unknown number to one side, and all other terms to the other side. Unite like terms, and. divide both sides by the coefficient of the unknown number. Exercise 40. Solve the following equations : 1. 12 X - 20 a: + 13 - 9 2^ - 259 ; 336 + (3 :c - 1 1) = 2 (5 2: - 5) + 8 (97 - 7 a:). 2. 62; + 4a; = 3:r + 84; 6a: + 2(13-2^) = 3 (17 -a;). SIMPLE EQUATIONS. 107 3. 2 (a; +2) + 182: = 3(5 + 2;) + 0; 30 a; + 20 2:- 15 a; + 12 a; = 2820. 4 9(2; _ 1) + 2 (a: -2) =10 (2 -a:); 2 (a: + 2) (a: - 4) = a;(2a;+ 1)-2L 5. 6y-2(9-4?/) + 3(52/-7) = 10 2/-(4 + 16y+35), and verify. 6. 2y-(4y-l) = 5y-(//+l); 56+ 21 a;- 8 (2 a;- 1) = 62. 7. 10 [224 -{x-V 192)] = 7 (28 + 3a:); 9 (7 + ^y) -4[9-(2-.y)] = 252y. 8 25 a:- V.) - [3-{4a:-3}] = a: — (a: — 5), and verify. 9 20(2~a;) + 3(a:-7)-2[a;+9-3{9-4(2-a:)}] = 1. 10. (y-2)(7-7/) + (7/-5)(y+3)-2(y^l) + 12 = 0. 11. 4 (v/ + 5)2 - (2 y + 1)2 = 3 (y - 5) + 180 ; 2.25 x - 1.25 = 3 a: + 3.75. 12. .15?/ + 1.575 -.875?/ = .0625 2^. Query. In transposing, why change the signs ? 49. Known Numbers are represented by the first letters of the alphabet, and by figures ; as, a, b, 2 c, 6. Unknown Numbers are usually represented by the last letters of tlie alpliabet; as, x, y, z. An Equation i.s a statement that two expressions repre- sent the same number. An Identical Equation, or an Identity, i.s one which is true for all values of the letters which enter into it ; as, {a + z) (a — a:) = a2 — 3^. 108 ELEMENTS OF ALGEBRA. The Roots of an equation are the values of the unknown numbers. The Degree of an equation is the power of the unknown number, and is determined by the greatest number of un- known factors in any term. Thus, X — y = 6 is an equation of the Jirst degree ; 4x^ + 5y = 3 and 5 xy + 2 — 3 X ave equations of the second degree. A Simple Equation is an equation of the first degree. Miscellaneous Exercise 41. Solve the following equations : 1. 5(7 + 3:?/)-(23/-3)(l-2?/)-(2,y-3)2 + (5 + 2/) - 0. 2. {22j+lf + {27/-lf=UyQf-4:)+57, and verify. 3. 1.5(26i/-51)-12{l-3y)=7Sy-2[5y-2.6{l-.Sy)]. 4. .6a:-.7:r + .752:-.875:r + 15 = 0; .6y-{.lSy-M) = .2 2/ + 4.45. 5. 30 ^ - 3 [30 z-{2z- 5)] = 5{2z- 57) - 50. 6. 10 (^ + 10) - 18 (3 2 - 4) + 5 (3^-2) (2 z - 3) = 30 2;^ — 16, and verify. 7. 4.Sy-2{.72y-M) = 1.6?/ + 8.9; .5x-.3x-.25 = .25a:-l. 8. .2 7/- .16?/ = .6 -.3; .5y - .2y = .3y- 15. 9. 5.6 y f .25 y+ .3y = y-'S; .6 y -\- .25 - A y = l.S -.75?/ -.3. 10. 3)x- .25 (x-2)~- .3 (3x-{- 12)? = 41. PROBLEMS LEADING TO SIMPLE EQUATIONS. 109 CHAPTER X. PROBLEMS LEADING TO SIMPLE EQUATIONS. 50. The beginner will find the model solutions of great benefit ill forming statements, and he should give them careful consideration before attempting to solve any of the problems in each set. Exercise 42. 1. A father is 35 and his sod 8 years old. In how many years will the father be just twice as old as the son ? Solution. Let x — the number of years required. Then x + 35 = the number of years in the father's age x years from now, and X -f 8 = the number of years in the son's age x years from now. By the conditions of the problem, at the expiration of x years twice the son's age, or 2 (x + 8), equals the father's age, or a: -f 35. Hence, the equation 2 (x + 8) = x + 35, or 2 x + 16 = x -I- 35. Transposing and uniting like terfns, x = 19. 2. One number exceeds another by 5, and their sum is 29. Find the numbers. 3. The difference of two numbers is 14, and their sum is 48. Find the numbers. 4. A father gave S200 to his five sons, which they are to divide according to their ages, so that each elder son shall receive $10 more than his next younger brother. Find the share of each. 110 ELEMENTS OF ALGEBRA. 5. A father is four times as old as his son ; in 24 years he will only be twice as old. Find their ages. 6. Divide 50 into two parts, so that three times the greater may exceed 100 by as much as 8 times the less falls short of 120. Solution. Let x = the greater part. Then 50 — x = the less part, and 3 X = three times the greater part ; also, 8 (50 - x) = eight times the less part. But, 3 X — 100 = the excess of three times the greater part over 100; also, 120—8 (50 — x) = the number that eight times the less lacks of 120. By the conditions, 3 a: - 100 = 120 - 8 (50 - x). Therefore, . x = 36, for the greater part, and 50 — a; = 14, for the less part. 7. Twenty-three times a certain number is as much above 14 as 16 is above seven times the number. Find the number. 8. A is five years older than B. In 15 years the sum of their ages will be three times the present age of A. Find the age of each. 9. A is 25 years older than B, and A's age is as much above 20 as B's is below 85. Find their ages. 10. The sum of the ages of A and B is 30 years, and five years hence A will be three times as old as B. Find their ages. 11. The difference between the squares of two consecu- tive numbers is 121. Find the numbers. PROBLExMS LEADUSG TO SIMPLE EQUATIONS. Ill Solution. Let x = the less number. Then will jc -f 1 = the greater number, x^ = the stjuare of the less number, and (x -I- 1)*^ = the stpuire of the greater number. Then (x + 1)^ - x'^ = the ditlerence of the scjuai-e numbers. But 121 = the difference of the squares. Hence, (x + l)^ - x- = 121. Therefore, x = 60, the less number, X -h 1 = 61, the greater number. 12. Find three consecutive numbers whose sum is 27. 13. The difference of two numbers is 3, and the differ- ence of their squares is 21. Find the numbers. 14. Find a number such that if 5, 15, and 35 be added to it, the product of the first and third results may be equal to the square of the second. 15. I sold a cow for S35 and half as much as I gave for it, and gained SIO. Find the cost of the cow. 16. A had four times as much money as B ; but, after, giving B $16, he had only two times as much as B. How much had each at first ? Solution. Let x = the number of dollars that B had at first. Then 4x = the number of dollars that A had at first. But 4x — 16 = the number of dollars that A had after giving B^16, and X + 16 = the number of dollars B had after receiving ^16 from A. By the conditions, 4 X- 16 = 2 (x + 16). Therefore, x = 24, the number of dollars that B had, and 4x = 96, the number of dollars that A had. 17. A father is 3 times as old as his son ; four years ago the father was 4 times as old as bis son then was. Find their ages. 112 elp:ments of algebra. 18. One number is two times another; but if 50 be subtracted from each, one will be three times the other. Find the numbers. 19. A has $26.20 and B has $35.80. B gave A a cer- tain sum; then A had four times as much as B. How much did A receive from B ? 20. If 288 be added to a certain number, the result will be equal to three times the excess of the number over 12. Find the number. 21. A farmer has grain worth $0.60 per bushel, and other grain worth $1.10 per bushel. How many bushels of each kind must be taken to make a mixture of 40 bushels worth $0.90 a bushel? Solution. Let X = the number of bushels required of the f 0.60 grain. Then 40 — a: = the number of bushels required of the $1.10 grain; and Y®^*\j X — the number of dollars in the cost of the $0.60 grain; also, 1.10(40-a:) — the number of dollars in the cost of the $1.10grain. Hence, ^-^^ a; -f- 1.10 (40 — x) — the number of dollars in the total cost of the mixture. But the cost of the mixture is to be $36. Hence, ^%x-\- 1.10 (40 -a:) = 36. Therefore, x = 16, the number of bushels of the $0.60 kind, and 40 - X = 24, the number of bushels of the $1.10 kind. 22. A merchant has two kinds of vinegar : one worth $0.35 a quart and the other $1.25 a gallon. From these he made a mixture of 63 gallons, worth $1.30 a gallon. How many gallons did he take of each kind ? 23. A merchant has a mixture of 88 pounds of 13 and 11 cent sugar, which he sells at 12| cents per pound. How many pounds of each kind are there ? PROBLEMS LEADING TO SIMPLE EQUATIONS. 113 24. I bought 24 pounds of tea of two different kinds, and paid for the whole $9. The better kind cost $0.65 per pound, and the poorer kind $0.35 per pound. How many pounds were there of each kind ? 25. A grocer having 75 pounds of tea worth $0.90 a pound, mixed with it so much tea at $0.50 a pound that the combined mixture was worth $0.80 a pound. How much did he add ? Remarks. No general method can be given for the solution of problems. ^ The beginner will find that his principal difficulty in solving a problem consists in forming the equation of conditions, and in order to overcome this, much will depend upon his skill and ingenuity. The statement of a problem consists in translating its conditions into algebraic symbols and ordinary language. Many times the be- ginner fails to form a correct statement, because he does not under- stand what is meant by the ordinary language of the problem. If he cannot assign a consistent meaning to the words, it will be impossi- ble for him to express their meaning in algebraic symbols. It often happens that the words appear to be susceptible of more than one meaning. In such caj^es the student should express the meaning that seems most reasonable in algebraic symbols, and obtain the result to which it will lead. Should such result be inadmissible, the student should Uy another meaning of the words. The student must depend upon hus own powers^ and should he at times be perplexed, he must not be discouraged, since nothing but patience and practice can overcome the difficulties and give him readiness and certainty in solving problems. He must study the iT»eaning of the language of the problem, to ascertain the unknown numbers in it. There may be several such numl^rs, but oftentimes a little skilful manipulation will enable one to express all of the un- known numbers in terms of some one of them. Select the one by which this can be most easily done and represent it by some one of the final letters of the alphabet. Among the following problems no doubt the beginner will find 8 114 ELEMENTS OF ALGEBRA. some which he can readily solve by arithmetic, or by guessing and trial; he may thus be led to undervalue the power of algebra, and to regard its aid as unnecessary. In reply, as the student advances lie will find that by the aid of algebi'a he can solve not only all of these problems, without any uncertainty or guessing, but those which would be exceedingly difficult, if not altogether impossible, if he depended upon arithmetical processes alone. 26. A's age is six times B's, and fifteen years hence A will be three times as old as B. Find their ages. 27. A is three times as old as B, and 12 years since he was fiv^ times as old. Find B's age. 28. A father has three sons; his age is 60, and the joint ages of the sons is 46. How long will it be before the joint ages of the sons will be equal to that of the fathei' ? 29. If yon walk 10 miles, then travel a certain distance by train, and then twice as far by coach, and the whole journey is 70 miles, how far will you travel by coach? 30. A is twice as old as B, and seven years ago their united ages amounted to as many years as now represent the age of A. Find their ages. 31. After 136 quarts had been drawn out of one of two equal casks, and 80 gallons out of the other, there remained just three times as much in one cask as in the other. Find the contents of each cask. 32. Find the number whose double increased by 1.2 exceeds 3.65 by as much as the number itself is less than 8.65. 33. Find three consecutive numbers such that if they be diminished by 10, 17, and 26, respectively, their sum will be 10. PROBLEMS LEADING TO SIMPLE EQUATIONS. 115 34. Two consecutive numbers are such that one fourth of the less exceeds one tifth of the greater by 1. Find the numbers. 35. There are two consecutive numbers such that one fifth of the greater exceeds one tenth of the less by 3. Find tliem. 36. Find a number such that the sum of its half and its fourth shall exceed the sum of its fifth and its tenth by 45. 37. Find a number such that the sum of its half and its fifth shall exceed the difiference of its fourth and its tenth by 110. 38. If a watch and chain are worth $185, and the watch lacks $19 of being worth two times the cost of the chain, find the cost of each. 39. If silk costs 6 times as much as linen, and I buy 22 yards of silk and 28 yards of linen at a cost of $52, find the cost of each per yard. 40. A man gave 17 boys $3.31, giving to some 13 cents each and to the rest 23 cents each. How many received 23 cents ? 41. I paid a bill of $1.53 with 39 pieces of money, some 3-cent and the rest 5-cent pieces. How many of each did it take ? 42. A son earns 37 cents per day less than his father, and in 8 days the father earns $6.08 more than the son earns in 5 days. Find the daily wages of each. 43. How many 10-cent pieces and how many 25-cent pieces must be taken so that 95 pieces shall make $12.35? 116 ELEMENTS OF ALGEBRA. 44. Divide $112 into two parts, so that the number of five-cent pieces in one may equal the number of three-cent pieces in the other. 45. A sum of money consists of dollars, twenty-five- cent pieces, and dimes, and amounts to $29.50. The number of coins is 55. There are twice as many dimes as quarters. How many are there of each kind ? 46. A sum of £8 17 s. is made up of 124 coins, consist- ing of florins and shillings. How many are there of each? 47. A bill of £4 5s. was paid in crowns, half-crowns, and shillings. The number of half-crowns used was four times the number of crowns and twice the number of shil- lings. How many were there of each ? 48. A bill of £48 J was paid with guineas and half- crowns, and 12 more half-crowns than guineas were used. How many were there of each ? 49. A company of 84 persons consists of men, women, and children. There are three times as many men as women, and five times as many women as children. How many are tliere of each ? 50. The sum of three numbers is 263. The first is 3 times the second, and the third is 23 more than 5 times the sum of the other two. Find the numbers. 51. A farmer wishes to mix 660 bushels of feed, con- taining oats, corn, rye, and barley, so that the mixture may contain two times as much corn as oats, three times as much rye as corn, and four times as much barley as rye. How many bushels of each should be used ? PROBLEMS LEADING TO SIMPLE EQUATIONS. 117 52. Divide $2590 into two sucli parts that the first at 7% simple interest for 8 years may amount to the same sum as the second in 5 years at 8 %. Note. The character % is sometimes used for the term "/>er cent." Per cent is used by ellipsis for rate per cent. Thus, au allowauce of 7 on a hundred is at a rate of .07, aud the rate per cent is 7. 53. $330 is invested in two parts, on one of which 15% is gained, and on the other 8 % is lost. The total amount returned from the investment is S345. Find the investment. 54. A man has $ 7585. He built a house, aud put tho rest out at simple interest for 18 months; 40% of it at 5 % and the remainder at 6 %. The income from both in- vestments is $211.26. Find the cost of the house. 55. In a certain weight of gunpowder the saltpetre was 4 pounds less than half the weight, the sulphur 5 pounds more than a fifth, and the charcoal 3 pounds more than a tenth. Find the number of pounds of each. 56. A company of 266 persons consists of men, women, aud children. Tlie men are 14 more in number than the women ; the children 34 more than the men and women together. How many are there of each ? 57. I bought 16 yards of cloth, and if I had bought one yard less for the same money, each yard would have cost $0.25 more. Find the cost per yard of the cloth. 68. A and B, 85 miles apart, set out at the same time to meet each other; A travels 5 miles an hour aud B 4 miles an hour. How far will each have travelled when they meet ? 118 ELEMENTS OF ALGEBRA. 59. $330 is loaned for nine months in two parts ; on one 15 % per annum is gained, and on the other 8 % per annum is lost. The total amount from the loan is $364.25. Find the amount in each loan. 60. A boy has a certain sum of money, lie borrowed as much more, and spent 12 cents; he again borrowed as much as he had left, and spent 12 cents ; again he bor- rowed* as much as he had left, and spent 12 cents ; after which he had nothing left. How much money had he at first? 61. A carriage, horse, and harness are worth $720. The carriage is worth eight tenths of the value of the horse, and the harness six tenths of the difference between the value of the horse and carriage. Find the value of each. 62. A boy sold half an apple more than half his apples. Again he sold half an apple more than half his remaining apples. A third time he repeated the process; and he had sold all his apples. How many apples had he ? Algebra is the science which treats of algebraic liumbers and the symbols of relation. Algebra, like arithmetic, is a science which treats of numbers. In arithmetic the numbers are positive and represented by figures. In algebra the letters of the alphabet or figures are used to represent numbers, and they may be positive or negative, real or imaginary. Algebra enables us to prove general theorems respecting numbers, and also to express those theorems briefly. FACTORING. 119 CHAPTER XL FACTORING. 51. A Factor is one of the makers of a number. Thus, since 5 with the aid of 4 and by the process of multiplica- tion makes 20, 5 is a factor of 20. A factor is also a divisor, but it is considered a divisor when it separates a number into parts, not when it helps to make up a number. Note. Unity cannot be a factor. Factoring is the process of separating an expression into its factors. Example. Find the factors of 12 a« 6 x^. Solution. The prime factors of 12 are 2, 2, and 3. The factors of a' are a, a, and a. The factors of x^ are x and a:*. Therefore, l2a*bxi = 2X2X3xaXaXaXbXxXxK Hence, a» a direct result of the principle that monomials are mul- tiplied by writing the several letters in connection, and giving each an exponent equal to the sum of the exponents of that letter in the factors. To Factor a MonomiaL Separate the letters into any number of factors, so that the sum of all the exponents of each factor shall make the exponent of that factor in the given expression ; also sepa- mte the numerical coefficient into its prime factors. Exercise 43. Separate into factors with integral exponents : 1. Ua^l^x; Wo^t/^; 15ah^(^; 20ab(^; Soa^f:!^^; 120 ELEMENTS OF ALGEBRA. Separate into two equal factors : 2. UaH'^; da^f; SI a^ b^ x^"" y^"" ; 169a"5. Eemove the factor 2 a^ bi from : 3. Sa^b; Qabx; IQab^c^; 10 a'H-^ x^y^. Separate into three factors, also into four : 4. cc ; m^ " ; a" ; xi ; x^. 52. Example 1. Factor a^x - Sa^x^. Solution. Dividing the expression by a^ x, we have a — 3 aj. Hence, a^ x — 3a^x = a'^x {a — 3 x). Example 2. Factor 5 a'^b^x^ - 15 ab^x^ + 20 b« x^. Solution. By examining the terms of the expression we find that 5 b^ x^ is a factor of every term. Dividing by this common fac- tor the other is fomid. Hence, the factors are 5 h^x^ and a^x — Sax + 4 6. .-. 5aH^x^-l5ab^x»-\-20b^x^ = 5b^x^(a^x-3ax + 4b). Hence, When the Terms of a Polynomial have a Monomial Factor. Divide each term of the expression by the common factor. The divisor and quotient will be the required factors. Exercise 44. Factor the following : 1. 7n^ + n; 4: a^b + ab^c+ Sab; Sa^- 12 a^. 2. ax — bx+cx;S9a^7/-{-57x^y^. 3. 5x^ + Sa^-x^; 72 h^ x'^y^ - 84:b^ x y^ - 9Q a b a^y^. 4. 924 «2 x^y^'z- 1 178 a x"" y z"" + 1232 a^ x"" y'^ z\ FACTORING. 121 5. 4:aH-(J0aI^+20abc-{-SaH*x^+Uahy-Z6aHcx^, 6. 2 xi y ^ a b X y + c a^ y^ ; 5 x^ + 10 x^ — 15 xi. 53. In certain Trinomiala, of the form x^ -\- ax -^ b, where a and b represent any numbers, either integral, fractional, positive, or negative, it is possible to reverse the operation of Art. 25, and sepa- rate the expression into the product of two binomial factors. Evi- dently the first term of each factor will be the square root of a;*, or x; and to obtain the second terms of the factors, /ind two numbers whose algebraic product is the last term, or b, and whose sum is the coefficient ofx, or a. Example 1. Factor x^ + 21 a: -{- 110. Solution. Evidently the first term of each factor will be x. The second term of the factors must be two numlnirs whose product is 110 (the third term), and whose sum in 21 (the coefficient of a;). The only two numbers whose product is 110 and whose sum is 21 are 10 and 11. Therefore, z« + 21 1+ 1 10 = (a: + 10) (x + U). Example 2. Factor x^ + x- 132. Solution. Evidently the first term of each binomial factor will be X. The second term of the two binomiul factors must be two numbers whose algebraic product is — 132 and whose sum is -f- 1 (the coefficient of x). The only two numbers whose product is — 132 and whose sum is -f 1 are 4- 12 and — 11. Therefore, x^ -\- x — 132 = (X -\- U) (x - U). Example 3. Factor y^ - 5cy — 50 c^ Solution. Evidently the first terra of each binomial factor will be y. The second term of the two binomial factors must Ijc two numbers whose product is — 50 c* and whose sum is —5c (the coef- ficient of y). The only two numbers whose pro<luct is — 50 c* and whose sum is —5c are + 5 c and — 10 c. .-. y^ — 5 cy — 50 c* = (y-l-5c)(y-10c). 122 ELEMENTS OF ALGEBRA. Example 4. Factor x^y^ — (jn — n) xij — m n. Solution. Evidently the first term of each binomial factor will ho, xy.. The second term of the two binomial factors must be two such numbers whose product is —nin and whose sum is — (in — n). The only two numbers whose product is — m n and whose sum is — (m — n) are + n and — m. .*. x^y'^ — (in — n) xy — m n = (x y -\- n) {xy — m). Hence, I. If the Coefficient of the Highest Power is Unity. For the first term of each factor take the square root of one term of the trino- mial ; and for the second term of the factors, such numbers that their algebraic product will be another term of the trinomial, and their sum multiplied by the first term of either factor will be the remaining term of the trinomial. Exercise 45. Factor the following : 1. ^24.19^+88; .^2-72^+ 12; ft8_20a4 + 96. 2. 2:2 + 35 a; + 216; 52(^2- 245c + 143. 3. ft* ¥ + 37 a2 62 + 300 ; a^ + 5 ah - 66 h\ 4. a?ly^-oah-24.- ft* + 15*2 + 44. a^ + 17 ft^ + 60. 5. a^y---dahc-lQc^; a^-2a'^-120; yi2+.8 ^i + l.o. 7. ^2 _ 15 a; + 44; ^„2 + .i_i. ,„, ^ i^_ . ^2 _ 11 ^^ _ 26. 8. 130 + 31 ft & + ft2 /;2 . ,,2 - 20 ah x+ 75 h'^ oj^ ; y^ + 6 a;2 7/2 _ 27 2)4; 1 + 13 ^ + 42 :r2 ; ^;i2 - 15 a m + 06 a^. 9. ft2 - lSax7j - 243 x^ y^; (x + yf + 5 {x +. ?/) + 4. 10. 40 ft2 2,2 _ 13 ,-^ 7, + 1 . (^ _ 5)2 + (^ -h)-2. FACTORING. 123 11. (x-yf-d{x-y)-l0;ay^+54x-]'729', 204-29^^+ .,4. 12. (a -hiy -^9 (a + i)2 + 8 ; 2;*" - (6 +'m) x"^"" -^ b vi. 13. a2 - 10 a V^v - 39 h^c^- x^- -^ {a - h) x"" - a b. 14. a^-9xij-70f', a;2-.|.c-|; 2;*''-43ic2n^46Q 15. 2^-{-iax--^^a^; x*-a^x'^-4e2a\ 16. x^i/-^Sx7/-154:; a^" x*"' + 14 a" x^"' 2/" + 33 y^. 17. a.>2 ?/2 _ 28 a" i" a;?/ + 187 a^- b^- - x^^\x + '^^V- 18. a;*"//" + 20a'"^/'";;t2''7/2n ^ 5irt2«.j2m. (^ ^ ^^^em -- 7 a*" (a; + ?/)3- - 98 a^" ; n* + .01 n^ - .011. 19. ^+^x-^\; u;2+2^2/~.21y2; «4+_8^^2+ j^. By an extension of the foregoing principles we may factor some trinoniials, of the form c^x^ + ax -\- hd^ where the coefficient of a:^ is a perfect square. Thus, Example 20. Factor 4 a:« + 4 a: -^ 3. Solution. The first term of each binomial factor will be the square root of 4 x^. The second term of the two binomial factors must >>e two numbers whose product is — .3 and whose sum multiplied by 2 a; is + 4 x. The only two numbers whose product is — 3 and whose sum multiplied by 2 a: is + 4 a; are + 3 and — 1 . .-. 4a:« + 4a;-3=(2x + .3)(2a:-l). 21. 4a.^-10:r + 6; 9a^»-27a;+18; 4a'2+16aa;+12a2 22. 9 rt2 + 30 « 5 + 24 J2; 16 a^« - 20 a a; + 6 «« 23. 25.xiO'"-|2r5"'rt''-Ja2-; 36(a-t)*- + 12(a-6)*" + * - 143 (a - h)\ 124 ELEMENTS OP ALGEBRA. 54. We may factor some trinomials of the form ax^ + bx ■{- c. Thus, Example 1. Factor 8 a;^ - 38 a;+ 35. Solution. The first term, 8x% is the product of the first terms of the binomial factors. The last term, 35, is the product of the second term of the two binomial factors. It is evident that the first term of each binomial factor might be ±2 a: and ±.4x, or ±Sx and ±a:; also the last terms of the two factors might be ± 7 and ± 5, or db 35 and ± 1. From these w-e must select those that will produce the middle term, —38 x, of the trinomial. Since (+2x) X (— 5) + (+ 4 cc) X (— V) = — 38 X, we must take + 2 a: and + 4 x for the first terms, and — 7 and — 5 for the corresponding second terms of the two bino- mial factors. Therefore, 8 a;^ - 38 a; + 35 = (2 a; - 7) (4 ic — 5). Example 2. Factor 6x* ~ bx'^y^ — 6 y\ Solution. Take + 3 a;^ and + 2 x^ for the factors of 6 x*, and + 2 y'^ and — 3 2/^ as those of — 6 ?/*. We now arrange them in bino- mial factors, so that the algebraic sum of their cross products shall be - 5 a:2 1/2. Since (+ 3 a^) X (- 3 7/2) + (+ 2 x^) X (+2y^) = -5x^ y\ + 3 a;2 and + 2 a;^ are the first terms, and + 2 y'^ and —3y^ are the corresponding second terms of the factors. .•. 6x^ — 6 x^y^ — 6y* = (3 a:2 + 2 ?/) (2 a:2 - 3 y^). Hence, II. If the Coefficient of the Highest Power is not Unity Arrange the trinomial in descending powers of a common letter. Select factors of the extreme terms and arrange them in binomial factors, so that the algebraic sum of their cross products shall be the second term of the trinomial. Exercise 46. Factor the following : 1. 4 x^ + l.S a: + 3; 4 2/2 _ 4 y _ 3; 12 a^ -\- a^a^ - x\ 2. S + llx-4:X^; Sx^'-22xy-21f; ^o?'x^ + ax-^l. FACTORING. 125 3. 8 m6 - 19 m3 _ 27; 15 a^ _ 58 a + 11 ; 6 a2 ^. ^ ah - 3 62; 2 //<2 _ 13 m 71 + 6 n2 ; 3 a^» + 7 a; + 4. 4 24 + 37a-72a2; 15^:24. 224a: -15; 4-5a;-6a:2. 5. 6a^»- 19.ry + 10/; % a^ -^^ U xy - Ibf) lb a^ -77 a; +10; 24 a,^ + 22 2;- 21 ; lla2 + 34a4- 3 6. 18-33a;+5ar'; Ga:2_7a,2^_3^. 5 + 32;«- 21a;2. 7. 24a,'2-29a;y-4y2; ea^^+lQ^ra-yn^y^^m 8. 2(a;+y)2+5(a;+y)(m + ?i)4-2(m + n)a; 2«2+a;-28. 9. 2(x+yf-l{x^y){a^h) + Z{a + hf- l^x'j^^x-^. 10. n{x-yf''-l'6x'^y^{x-ijf''^2x^'^f) 27a2+6a-l. 11. 8 a2" -f 34 rt» (2: - yY^ + 21 (a: - ;/)2'»». 55. A trinomial is a perfect square when two of its terms are positive, and the third term is twice the product of their .square roots. Such trinomials are particular forms of I., and their binomial factors are equal. Example. Factor 4 z^ + 44 a:y + 121 y\ Solution. The first term of each binomial factor will be the 8C[uare root of 4 x^, or 2 a: and 2x, For the second terms of the bino- mial factors tjike the square root of 121 ?/*, or 11 ?/ and 11 y. Since the terms of the trinomial are positive the factors are 2 x + 1 1 y and 2x + lly. Therefore, 4x« + 44xy4- 121 y2= (2x+lly)(2x+ 11 y) = (2x4-lly)^. Hence, III. If the Trinomial is a Perfect Square. Arrange the tri- nomial .according to the powers of one letter. For one of the equal factors, fin<l the square roots of the first and last terms, and connect these roots by the sign of the second term. 126 ELEMENTS OF ALGEBRA. Exercise 47. Factor the following : + 225&2c2; a6-4a4 + 4a2 2. 49 m6- 140 m%2 +100714; si x"^ 9/ -126 a^xhj + 49 a^. 3. 7n}^—2m^ni-n'^; l-10mn + 25m^n^; x'^+2x^i-x^. 4. {a + hf + 16 (a + ^>) + 64 ; 7?i2 + 18 m + 81. 5. 4a*a^-20a2^7/ + 25a;42/^ o61aH^c^-16ahcdmn + 4:d^m^'n?; 121 7/2,27^4 - 220 mn'^2^ + 100^2 6. 225 X^ - 30 a;2 7/2 + ^4 . 4 ^4« _ 4 ^2n ^m _^ ^2m^ 7. 49 m27i2 + -2^ m n^ + ^71*; ^j2 + ^ + 1. 8. -^^a^ + ^^^W + \a^h^- a^c + 6a^h^e+9Wc. 9. 9^:2 _ 3:^7/ + ^7/2; (m - 7O2 + 2 (7?z, - ^i) + 1. 10. {ci?-af-\.6{a^-a)+ 9; 4 (a; + 7/)2 + Jg- + a? + 7/. 11. at + 6l — 2 aHf ; 7?i — 2 ??ii + 1 ; ??z2 71 + 77^ 7i2 — 2 77it 77!. 12. x-\-2 x^y^ + 7/ ; 7?i2 n + a2— 2 a m n^ ; 4 ic+ 1 2 71 a:^ + 9 7i2. 13. (a + 2>)2"-10(a + &)"c + 25c2; | ^5m_^ _i_6___ 11 J.«. 56. Example. Factor 8 a;^ - 27 i/^. Solution. Evidently (Art. 34) 2 re - 3 y is a divisor of 8 x^ - 27 1/^. Dividing 8a:^ l)y2x, we have 4x^, the first term of the quotient. Divide 4 rc^ by 2 oj, multiply the result by 2y, and we have Qxy, the second term in the quotient. In like manner we find 9 .y^ for the last term in the quotient. Hence, the quotient is ^x^ + Qxy + ^ y^. Therefore, the factors of the binomial are 2ic — 3?/ and 4x'^+ 6a:?/ + 9?/2. FACTORING. 127 Since the dividend is equal to the prcKluct of the divisor and quotient, :i* — 27 y* = (2 X — 3 y) {4 x^ -h 6 xy + d y^). Hence, in general, When a Binomial is the Difference of Two Equal Odd Powers of Two Numbers. Cunsider the binomiid a dividend, and find a divisor and quotient by inspection (Art. 34) The divisor and quotient will be the required factors. Exercise 48. Factor the following : 1. l-d'i'Sa^; 82^-7297/) 216x^-a^ 2. 2^,/ - aH^; x^ -1; 243 a^ - b^; a^h^ - m^ 3. 216 d? — 343 ; 3 a; — 81 a:*. Suggestion. Remove the monomial factor 3x first. 4 a}^ - 1024 6^0 ; 729 x^ - 1728 f\x-^- y-\ 5. 135a;*-320ar2; 2an-64a&; a;-^-7/-f. 6. a655-2:6/; 64^6-125^3; a:3n_^« 57. Example. Factor 729 + a«. Solution. Since 729 is the 6th power of 3, 3* + r|2 (Art. 36) is a divisor of 729 + a'- Dividing 729 by 3^* we have 3*, the fii-st term in the quotient. Divide 3* by 3', multiply the result by a'*, and we have 3* a*, the ."^econd term in the quotient. In like manner we find a* for the last term in the quotient. Honcc, the quotient is 3* - 3"^a2 + n*. Therefore, the factors of the binomial are 3* -f a* and 3* — 3^ a-* + a*. Since the dividend is efpial to the ]>roduct of the divisor and quotient, 729 + (i« = (9 + a^) (81 - 9 a* + a<). Hence, in general, When a Binomial is the Sum of Two Equal Odd Powers of Two Numbers. Let the student supply the method (See Art. 36). 128. ELEMENTS OF ALGEBRA. Exercise 49. Factor the following : 1. d2a^ + 1; 1 + x^; a^ + y^; x^^ + i/^. 2. a^ + 128; x^ + 729 ^/^ 64 2^6 + y^ 3. aH^ + 2^10^10; x^ +64/; 1000 a;3 + 1331 ^/^ 4. a^l8 + yS J 135^5 _!_ 320 :i;2. ^24 _|. ^24, 5. 2;-5 + 7/-5; j;15 + ^6. ^5^5 +^5^5. ^21 ^ J54, 6. a54 + 654. 1 4. ^12. ^^n _|_ ^6m. ^-f _^ ^- f ^ 7. ai2» + &9'" . 32 rj> h^ c^ + 243 a^ ; 1024 a^ + h^^. 8. 64 a;6 + 729 a^ ; yig a^ + g\ je ; (^2 _ J c)^ + 8 h^c^. 58. Example 1. Factor 25 a;^ - 64 2/*. Solution. The square root of the first term is 5 x^, and of the last term 8 y. Hence, since the difference of the squares of two num- bers is equal to the product of the sum and difference of the numbers (Art. 26), 25 a;2 - 64 3/2 = (5 x + 8 ?/) (5 a; - 8 y). Example 2. Factor {6 a - 4)^ - (3a + 4x - 4)2. Solution. The square root of each term of the binomial is 5 a — 4 and 3 a + 4 ic — 4. Adding the results for the first factor, we have 8 a 4- 4 X — 8, or 4 (2 a + a; — 2). Subtracting the second result from the first for the second factor, we have 2a — 4x, ov 2 (a — 2 a:). Hence, the factors are 4 (2 a + a: — 2) and 2 (a — 2x). Process. (5a-4)2-(3a-f-4a;-4)2 = [(5 a-4)+(3 a-|-4 .'c-4)] [(5 a-4)-(3 a+4 x-4)] = [5a-4 + 3a + 4a;-4][5a-4-3a-4a; + 4] = [8a + 4a;-8][2a-4a:] = 4[2a + a:-2][2(a-2a:)] = 8(2a + a:-2) (a-2x). FACTORING. 129 A binomial expressing the diiference between two e<iUttl even powere of two numbers may often be separated into several factors. Thus, Example 3. Factor a" — i/". Solution. The square root of each term of the binomial is 3* and y*. Adding these results for the first factor, we have x^ -}- y^. Sub- tracting the second result from the first for the second factor, we have x^ — y*. Similarly the factors of x* — y* are ar* + y* and x* — y^. In the same way the factors of z* — //* aie x^ + y^ and x^ — y"^. Finally the factors of x^ — j/* are x + y and x — y. Hence, the factors of the binomial are x* + y*, a:* -I- y*, x* + y\ x + y, and x — y. Process, x^' — y^^ = (x^ + y^) (x^ — y^) = (x^+y^)(x^ + y*)(x*-y*) = (x^ + y^) (a:* + /) (x2 + y2) (x2 -1/2) = (a:«+/) {x*i-y*) (xH/) (x + y) (x-y). Hence, in general, When a Binomial is the Difference of Two Equal Even Powers of Two "Numbers. Find tlie square root of eacli term of the biuouiial ; add the results for one factor, and subtract the second result from the first for the other. Notes : 1. Tlie preceding method is a direct consequence of Art. 26. 2. The above method finds a practical application when it is necessary to find the difference l)etween the squares of two numerical numbers. Thus, (235)« - '219)2 = (235 + 219) (235 - 219) = 454 X 16 = 7264. Exercise 50. Factor the following : 1. a^x^-lP- f ; 10 .r2 - ^y2 . 25 a^x^ - 49 JV- 2. a,-* - ?/ ; 2r4 - 81 / ; .x-^ - ?/ ; x^^ - f. 3. a86*-81a^?/«; l-100aH*c2; 16 a^^ - 9 6«. 4. 9 «2'. - 4 a^- ; i «2 - J /,2 . .^1 __ yl^ 9 130 ELEMENTS OF ALGEBRA. 5. x-^ - /; (^ + hf - (c + df- {X - yf - a\ 6. a^-[x-yf'Axy^a hf -I) {a ■\- hf - {a -- hf. 7. (a + l)2-(&+l)2;Xa+ir— C^^-l)2; (753)2 -(253)2. 8. (24 X + yf -{2Sx- yf; (1811)2 _ (689)2. 9. {5x- 2)2 -(x- 4)2; (1639)2 - (269)2. 10. a^"- 1; 729x'^y- xy^; aH - b^ ; a - h. 11. (2x+ a- Sy -(3-2xf; 64:X-^ - 729 y-^ 12. (575)2 - (425)2 ; 2 a - 4 2:2 ; 25 a" - 3 6''" ; a:^ - 3/6 59. Compound expressions can often be expressed as the differ- ence of two equal even powers of two numbers, and then factored by the foregoing principles. In many such expressions it will be neces- sary to rearrange, group, and factor the terms separately. Thus, Example L Factor x^ - y"^ -i- a^ - b^ + 2 ax - 2h y. Process. *2_y«-|-a2-6H2ar-2&i/=:a:2+ 2 ax + a^ - ¥ - 2hy - y^ = (x2 + 2 a a: + a2) - (62 + 2 6 y + t/2) = (X + a)2 -(b + yy = [(X + a) + (^ + b)] [(x + a)-(h + y)] = [a + b -i- X + y][a — b + X - y'\ Explanation. Rearranging and grouping the terms, in order to form the difference of two perfect squares, we have the third expres- sion. Factoring the third expression gives the fourth expression. The square root of each term of the fourth expression is (x + a) and (y + b). Adding these results for the first factor, we have a + b + X + y. Subtracting the second result from the first, we have a — b + X — y, for the second factor. FACTORING. 131 Example 2 Factor 2xy + I - x^ — y^. Process. 2zy + I - x"^ - y* = I - x^ + 2 xy - y^ = 1 -(x2- 2x1/4-2/2) = 1- (x-yY Art 55. = [l + (^-2/)][l-(^-l/)] = [l + x-y][l-x + yl Example 3 Factor 4a^b^i-4c^d^^8ahcd-(a^-\^b^-c^-(P)'' Process. 4 a* />» + 4 c2 </« + 8 a 6 c </ - (a^ + />' - c^ - d^y = 4a^b'^ -\-8abcd + 4 c^d^ - (a« + b^ - c^ - d'^y = (2a6 + 2crf)2- (a2 -I- 62 - c2 - rf2)a = [(2a6 + 2c</) + (a2-f-6*-c2~rf''')][(2aft+2c(/)-(a2 + 62-c2-f7i)] = [2a6 -f- 2cd + a* + fc'* - c* - «/2J [2a6 + 2c</ - a^ - &« + c^ + d^] = [(a2 + 2a6 + 6-')-(c^-2crf + rf2)][(c2+2crf + f/2)_(rt2_2a6+/^'^)J = [(a + ^y - (c - rf)2] [(c + dy - {a - 6)2] = [(a+6) + (c-rf)] [(a+6) - (c-^)] [(c+rf) + (a-6)] [(c+d)-(a-6)] = [a + 6 + c - (/] [a + 6 - c + dj [a - 6 -}- c + </] [6 + c + rf - a]. Explanation. Arranging and factoring the first three terms, we have the third expres.sion. The f*quare root of each term of the third expression is 2ab + 2cd and a"^ + 1^ — c^ — d^ Adding and subtracting these results, respectively, gives the fifth expreasion. Rearranging (2a 6 and 2 erf suggest the proper anangenient) and grouping the.se terms, gives the sixth expression. Factoring the terms of the sixth expression, we have the seventh expression Finally, factoring the terms of the seventh expression, we obtain the result Exercise 51. Factor the following expressions : 1. a2-62-c2-2&c; a^ -{- y'^ -V^ -2 a y • l^-a^-V^^2ah. 2. 25a:2_2^_(56c_9c2; 2?Jf.2ax-\- a^- y^- 2yz-z\ 3. 4a;2_i2a:y+ 97/2-81; T^-^x^Uf, 4r*-l+r)r-9z2. 4. 16^*- r2-f - - J 9rt2_6^4-l-a:2_8a;y-16y2. 132 ELEMENTS OF ALGEBRA. 5. x^ — li^ + '^'^^ — n^— 2mx — 2ny\ a* + b"^ — c^ — cf^ 6. 12 xrj -4 x^ -^J y^ + z^; A x - 1 - 4= x^ + 4 a\ 7. (^2 _ 2/2 _ ^2)2 _ 4 2/2;22 . (^2« ^ ^2" _ c4m)2 _ 4 ^,2«52« 8. 4 ^2 _ 12 a a; - 6-2 - fZ2 - 2 c c^ + 9 a2 . 4 a:^ + 9 ?/2 -16^2 -25 6^2 - 12^?/ - 40 ^^s; Ax^ - W' -2hc-c\ 9. :i'4 - 25 «« + 8 a2a;2 - 9 + 30 a^ + 16 a* ; 2/2 + 6 6aj - 9 62^2 _ 10 ^) ?/ - 1 + 25 ^>2 ; (a4« - 4 a2" - 6)2 - 36. 10. rz;2« _ 9 ^2 + ^2- _ 2 ^"^"^ _ 6 a & - ^>2. 11. a;6"-4?/*'" + 12?/2'"^ + 2a3:i.^«_ 9:^2 4. ^6_ 12. 4 ^2 _ 9 2/2 + 16 ^2 _ 36 ^2 _ 16 ^ ^ + 36 ny. 13. tt2« -f- ^>2« _ 2 a*^^)'* - c2'» ^ Aj^"* - 2 c"* A;2m^ 14. 4 ^2 + 9 ^.2 _ 16 (2/2 + 4 ^2) _ 4 (16 2/;2 _ 3 «^.). 15. a2_,_^5_9^,2 4.i^2. a^-a2-9-2a2^,2 + ^4_^6a. 60. The method for factoring a trinomial consisting of two trino- mial factors depends upon the following axiom : 5. If the same numher he both added to and subtracted from another^ the value of the latter will not be changed. Example 1. Factor x^ + a^x^ i- a*. Solution. Adding and subtracting a^x"^, we have x* + 2a^x^i-a* - a^x^. Factoring the first three terms of this expression, we get, {x^ + a2)2 — a^x^. Here we have the difference of two equal even powers of two expressions, and it is equal to the product of the sum and difference of their square roots. Hence, the factors are a* + a a: + a:^ and a^ — ax + x^ FACTORING. 133 Process. = X* i- 2a2x2 + a*-a«z« = {x^ + a^y-a^x* = {xHa^-a x) (xHa'^-a x), or (aHa x+x«) (a«-a z+z«). Example 2. Factor 16 a* - 17 a" 6^ + 6*. \ Process. I6a*-na^b^-^b* = \6a*-n a^b^+9a^b^-hh*-9a^b^ = 16a<-8a«fc2 + M-9a262 = (4 a^- 62)2- (3 a 6)-^ = (4a2 + 3a6-6«)(4a2-3a6-Z>2) = (rt + 6) (4 a - 6) (a - ^) (4 a - 6)v Explanation Adding and subtracting 9 a* 6* to the expression (to form a perfect square), arranging and factoring the terms, we have the fourth expression (the difference of two equal even powers). Factoring the fourth expression, we get the fifth expression. The factors of 4 a' + 3 a 6 - 6* are a + A and 4a - b. The factors of 4 a* — 3 a ^ — ^^ are a — b and 4 a + b. Hence, When a Trinomial is the Prodnct of Two Trinomial Factors. Make the trinomial a perfect scjuare by adding the requisite expres- sion. Also indicate the subtraction of the same expression. The resulting expre8.sion will be the difference of two squares. Take the sum of their scjuare roots for one factor, and their diflerence for the other. Exercise 62. Factor the following expressions : 1. 9a* + 3a2624.4fe*; aHOa^ + Sl; 16 X* -\- 4 aP f ■\- 1/*. 2. a^ + a^iZ-^-f; Sla^2Sa^a^-\-Ua^; mHm^wa + w*. 3. 40^+8^*2/^+9?/*; a8 + a*fe2 + 54. 81a* + 36 aHie. •1 25a^-9a^}^-\'l6b^; x^ + xy-hy^; a^ + a^f + f. 5. 16a8 + 8a*&3+92>«; 9a*'{'38aH^+49b^; p^ + pHh 134 ELEMENTS OF ALGEBRA. 6. 49 a^ + 110 a?lP' + 81 &^ 9 ^* + 21 ^2 ^^ + 25 /. 7. m*" + m2» + 1 ; ^*" + 16 :2;2« + 256. 8. a2-3a&+&2; «4«_6^2nj2m_^j4m. 25m4-44mV+16?i*. 61. Frequently the terms of an expression can be grouped so as to show a common factor. Thus, Example 1 . Factor 2am + ^hm — cm — A an— Qhn + 2 en. Process. 2am + 36m-cm — 4an — 66n + 2cn = (2 a m — 4 a 7i) 4- (3 & m — 6 6 ?i) — (cm — 2cn) = 2a (m— 2n) -h Sb(m — 2n) — c (m — 2n) = (m - 2 n) (2 a + 3 6 - c). Explanation. Grouping the terms of the given expression in pairs ; taking the common factor 2 a out of the first, 3 b out of the second, and c out of the third, we have the third expression. Divid- ing the third expression by m — 2n (the common factor), we have 2 a 4- 3 & — c. Hence, the factors are m — 2n and 2 a + 3 6 — c. Example 2. Factor 12 a^ - 4 a^b - S a x^ + b x^. Process. 12a^- 4 a^b -3 ax^ + bx^ = (12 a^-Sax^) - (4a^b-b x^) = 3 a (4a'^ - x^) - h (4 a^ - x^) = (4 a2 - x^) (3a-b) = l2a + x)(2a- x) (3 a- b). Explanation. Grouping the terms in pairs ; taking the factor 3 a out of the first, and b out of the second, we get the third expres- sion. Dividing this by 4 a^ — x^, we have 3a — b. The factors of 4a^— x^ are 2 a + x and 2a — x. Hence, the factors of the poly- nomial are 2a + x, 2a — x, and 3 a — b. Example 3. Factor 2mn — 2nx — my + xy -{- 2 n^ — ny. Process. 2mn — 2nx — my + xy + 2n^ — ny = (2mn — 2nx + 2 n^) — (my — xy + 7iy) , = 2n (m — X + n) — y (m — X + n) = (m — X + n) (2 n — y). FACTORING. 136 Example 4. Factor -4ax + 4x^ + 4ay-\-4y^-8xy. Process. —4ax-^4x^4ay+4y^-6xy = 4[-ax+x^-\-ay+y^-2xy] = 4[{x*-2xy-{-y^)-{ax-ay)] = 4[{x-y){x-y)-a{x-y)] = 4(x-y)[(x^y)-a] = 4{x-y)[x-y-a]. Example 5. Factor 2am^-2an^-2am-2an-\-2a^2a^4a^n. Solution. Remonng the common factor 2 a, we have w* — n* -7n — n + a — a*4-2an. Arrange the terms of this expression into the groups m* — (n* — 2 a n + a*), and — (m + n — a). The factors of the first group are m + n — a and m — n + a. Hence, m^ — n' — m — n + a — a2-|-2an = m*— (n* — 2an4-a*) — (m + n — a) = (m + « — a) (m — n + a) — (m + n — a). Dividing this expres- sion by the common factor, m + n — a, we have m — n + a — 1. Hence, the factors of the polynomial are 2 a, m + n — a^ and m-n-\-a- 1. Therefore, 2am^-2an^-2am-2an'{-2a^ ~ 2 a* -h 4 a^ n = 2 a (m + n - a) (m - n + a — I). Process. 2 am^ - 2 a n^ - 2 am - 2 an + 2 a^ - 2 a* + 4 a^n = 2 a[m^ - n^ - m - n + a - a^ + 2 an] = 2a [(my - (n - a)^ -(m + n- a)] = 2a[(m + n - a) (m — n i- a) — (m + n - a)] =: 2 a (m -f n — a) [m — n + « — 1] . Hence, To Factor a Polynomial by Gronping its Terms. Group the terms of the polynomial so that each group shall contain the same compound factor. Factor each group and divide the result by the compound factor. The divisor and quotient will be the required factors. If the polynomial has a common simple factor, remove it first. Note. It is immaterial what terms are taken for the different groups so that each group contains a common factor. Tf the groups are suitably chosen the result will always be the same, although the order of the factors may be changed. Thus, in Example 3, by a different grouping of the terms, we have 2mn — 2nx-my + xi/ + 2n^ — ny = (2mn — my) — (2nz — xy)-{-(2n* — ny) = w (2 n - y) - X (2 n - y) + n (2n - y) = (2n-y)(m-x + n). 136 ELEMENTS OF ALGEBRA. Exercise 63. Factor the following : 1. a^ ■{- ab -\- ac + be; a^c^ + acd — '2abc — 2bd. 2. am—bm—an+bn; 4:ax—ay—4:bx+by; af^+a^-{-cr-\-a. 3. Qax — Sbx — Qay + 3by; pr -\- qr — 2^ s — qs. 4. ax— 2hx-\-2by-\-4tGy — 4:cx— ay. 5. 5a2- 5&2_2a + 2&; ^ x^ + Z xy - 2 ax - ay. 6. 2x^-3^^-4cX-2; a'^x^-a^x^-a'^x^+1; mx-2my — nx + 2 7iy\ 4: X — a X -\- 4: a — a^. 7. x^ + mxy—4:xy — 4:my'^\ 4:a^ + 4:x'^ + 5a — 5x — Sax. 8. 3a'^—Sac — ab + bc; a'^ x + ab x + a c + ab y + I'^y -{- b c. 9. ^ ax^ + 3 a xy—5bxy—'Sby'^; mn + np — mp — n^. 10. rii^ + np — mf — 7i?\ V^y^—2 x^y + ?>a^—21 xy^. 11. 21 a- 5c + 3 «c- 2&C- 14 &- 35; i^2_5^2/ + 62/^+32: — 6?/; 2:^ — r?;2-|-2;_l, 62. Example L Factor a;^ + 1/2 + g'-J - 2 a; 1/ + 2 a; 2 - 2 ?/ 2. Solution. The expression consists of three squares and three double products. Hence, it is the square of a trinomial which has x, y, and z for its terms. Since the sign of 2 x 2 is + , and 2xyis—, X and z have like signs, while x and y have unlike signs. Hence, one of the two equal factors is x — y + z. .-. x^ + y^ + z"^ - 2 xy + 2 xz - 2yz= (x - y + z)^. Example 2. Factor x^ - 3 x^ y -\- 3 x y^ - y^. Solution. It is seen at a glance that the given polynomial fulfils the laws stated in Art. 29. Therefore, one of the three equal factors is x - y. .'. x^ — 3 x^ y + 3 X y^ — y^ = (x — yy. Hence, FACTORING. 137 When a Polynomial is a Perfect Power of an Expression. By observing the exponents, coetticients, and signs of the terms, find such expression, as raised to a given power, will produce the polyno- mial. This expression will be one of the equal factors. Exercise 64. Factor the following : 1. a^ -f 2 a 5 + Z>2 + 2 a 6- + 2 6 c + c2 2. a2-2a6 + 62_2«c-i-26c + c2. 3. a^+h^-\-c^-\-2ab-2ac-2bc; 16-f 32a: + 24ic2 4. a^-15a*x + 90a^x^- 2433:^ - 270a^JT'^+ 405aa^. 5. a^-2ab + }r^+2ac + c^-2ad-2bc + (l^-2cd-i-2bd. 6. 27x^i/-l08a?x^y^-64:a^+lUa^xy, 7. m^—2p x—2 n x-\-n^+]f—2 mn+2m x-\-3?—2 mp+2 np. Miscellaneous Exercise 55. Factor the following : Vott. If the expression has a common simple factor, it should be first removed. 1. 10ar»"-30a:"-40; ar^ + A^+l; \22^if-Z(Sxy-A8. 2. a;2 _ .56 a; + .03 ; a2 + f I a + 1 ; ^ ^ x - x^. 3. 3m«n8-3m*7i; 16^8-2; a^-Sl; 6a;54-48aJ*+72a^. 4. ar»ya-5^^y-^j; aH2-|^a3J-^; 9(a + J)2- + ^xy{a + by-x^f. 138 ELEMENTS OF ALGEBRA. 5. a'^—2ax—4:a + x^+4:x; 8— 2x^—4:0^— 2x^; da^+a. 6. x^ + if ^^ - 3%; ^2« + 16 a" + 63; -i-| a^'" + (f ^t" - -¥- ^^") a^'" - 6 a** +^. 7. m^ — a m — 71 m + a 71 ; a^ + 7 a — 8 ; 4 a^ — 4 2?^ -2a + 2b; 49 a^H^"" + 7 (:i;2 + 32/^)a"53«^i _^ 3iz;i2/i 8. 204-5a-a2; ^8" - -L2_8 ^4n ^ 1|^ 9. m^ — n^ — mp — np ; a;^ — ic^ + a;* — 2^^ ; a^ — a^ &^. 10. 380-^-^2. 8 ^10". _ 9 ^5m _|. 1 . (^^ _ ^)4« 11. 6a.^2_2,__ 77. 12 2:2+108^+168; ^2^2^^^/ + 2/2 — 5^—5?/; I 2^2 — Jg- (5 -m 71 + 3 2/) ^ + ^z'^^^'^ V- 12. l-Tiy^-Tfy^^ 2 2^2+ 5 ^?/- 3?/2-4aa; + 2a7/. 13. a2« + (^ _j_ ^) ^«^« + ^/2" ^7/; :r2 + (a + &) ^ - 2 a2 - a & ; a;- 12 - 7/^ ;. 81 x^ - 22 2^2 ^2 + y^^ 14. al2& + Z;13. ^3 + ^3_^3^^(^_|_^). a:4'»+c(a + 5)2:2« — ah {a — c) (b + c) ] m^ + 71^ + 771 + 71. 15. a2_2,2_c24.^/2_2(^^_2,c); 4 + 4^'+2«y+^2_^2_2^2^ 16. x^-V{a + 2h)x-\-ah-^h'^; x^^''+(a-b)x^''-2a'^-2ah. 17. 250 (m -nf±2; 8 (771 + nf ± (2 m - n^. 18. 52»c2»2;2'»_ 6"c"a2x" -&~C"2:"+ft2; 49^4_i5^2^2 + 121 2* ; (7?i + nf ± (m. - nf. 19. 4 (771 •— n)^ — (m — n); {m -{■ Tif ± m {m + n). FACTORING. 139 20. x^-b{a-c)x-ac{a-\-b)(b + c); 64:m^-{-12Sm^n^ 21. 6 2^ + rS3^i/ + 6xf- 6x^f-xi/-12f. 22. 2^'* -{She + ac + ab)x'' i- 'Sabc(b ■{■ c). 23. m3 + 4 m /i2 ± 8 n^ ± 2 m^ n ; {711 + 3 n)2 - 9 (m -pf. 24. x^'' + (a -\- b -c)x^^''- ac - be; {x -^ yf - x - y - 6 ; 25 ^ + 24 x^i/ + 16 f, 25. 9a:*y*— 3 2^^—60,-2 y^; m^— mri — 6 71^^47/1 ip 12 7^. 26. rrAn^{ab-zf-m^n^{xy^-2zf\ 81a2"_i99a»fe'» + 121 62-; 81 a^" - 99 a2«^,4« 4. 25 fts™ 27. 18 a:2 _ 24 xy + 8 / ± 36 a; ^ 24 v/ ; 2 m2 j^ 2mn - 12 ?i2 - 12 am - 36 a 71 ; a^a^ ± 64 a?iA 28. 2^^+ 3x'y-282/*+ 28y + 4a:; 2y - ^ay -\- 4.bx -\- G a X — 2x —4by. 29. 7W* 71 — 7?l2?i3 — 7;i3^2 _|_ ^^^ ,^4. ^j^4 _ ^^^^ ^ ^^4 30. 15 a^J - 16 7/2 - 15 a a: - 8 a; y + 20 a y ; (a - 6) (a2 _ c2) - (a - c) (a2 - ^2). 31. c«^-c2-a2c3rf3 4.rt2; m87l3±512;247n,27l2-3077l7l3 — 36 71*; aa^ — 3bxy — axy-\-3b/. 32. 7m2- 7n7i- 69124. 16m -327i; 4 a;^ + 4 ajS - a;2 - ar*. 33. 4 m8 - 4 7i3 _ 3 71 (712 _ m^) + 2 7?i (71 - 7?i)2 34 9m»±9a2m7; a.^~16y2 + a;±42^: (x-2xy)^ — (a; — 2 a; y) — 6. Query. How many factors in the first part ? 140 ELEMENTS OF ALGEBRA. 35. 64.(4: x + yf - 49 (2 a; - 3 7/)2 ; (wt* - iii^ - 5)2 - 25. 36. m^ + m^ 7i + m 7i^ + iii^ ii^ + m^ n^ + rn? n^ ; (a; — y)^ - 1 + 2:y (a; - ?/ + 1) ; (^2 + 4)2-16 a:2. 37. (m2 + 3 m)2 - 14 (m2 + 3 ??0 + 40 ; (m 7i - t!-)^ - m ?i (tw % — 72. — 3) — 9. 38. :C2H _ ^n _ I ^ ^-n ^ ^-2n. ^-f _ ^-f^ 39. 14^2 ^3_ 35a3 2;2+ 14a*:r; a;-6 -t/"! 40. 12:^:5-8^3^2+ 21 ^2^/; 64:cl± 27 ici Separate into four factors : 41. {x-2y)o(^-{y-2 x)t/; (^'"+ 6^"* + 7)2- (^'"+ 3)2. 42. 4 ^2 (^3 + 18 a &2) _ (32 a^ + 9 ^2 a?) ■ iQ ^^ ,^2 - (m2 + 4 ?^2 - ^2)2 . (^4m _ 2 rt2-^,2n _ J4«)2 _ 4 ^4m j4» ■ 43. x^ + ^./ - 8 0^6 ^3 _ g ^9 . ^9m _^ ^sm + 54 _^3»t + 64 44. m^ — 2 (?t2 + ^2^ ^.^2 _|. (^2 _ ^2^2 Separate into five factors : 45. m'' — inP n^+ 2 7?i* n^ —m^n'^; 6 m* 7^2 + m^ n — 6 m^ n^ -rn^n^l (a;2m _^ ^2n _ 20)2 _ (^^'^^n _ y2n _^ ]^2)2 46. ^7'« + ^4/n_;j^g^m__]^g. 16 ^7m_81 ^ »t_ ^g ^4m_^ 3]^ Separate into seven factors : 47, ^12m_^8mj4n_^4mj8n_^ J12n^ HIGHEST COMMON FACTOR 141 CHAPTER XII. HIGHEST COMMON FACTOR. 63. The product of any of the factors of a number is a factor of the given number. Thu8, since 30 = 2 X 3 X 5, 6, 10, and 15 are factors of 30. The product of the common factors of two or more num- bers must be a factor of each. Thus, since 42 = 2 X 3 X 7, and 66 = 2 X 3 X 11, 2 X 3, or 6, is a factor of 42 and 66. The product of the higliest powers of all the factors which are common to two or more numbers must be the greatest common factor of the given numbers. Thus, since 24 = 2» X 3 and 36 = 2^ X 3^, 2^ X 3, or 12, is the greatest common factor of 24 and 36. The Highest Common Factor (H. C. F.) of two or more algebraic expressions is the expression of highest degree which will divide each of them exactly. Thus, 3a:«y«i8the H.C.F. of 3a:«j/«, Qx^y\ and 15x<y«2. Note 1. Two or more exprcRsions are said to be prime to each other when they have no common factor. Thus, 5 a* and 9 b are prime to each other. Example 1. Find the H. C. F. of 24 a« 6« c«, 60 a'^'c^y^ 48a»62c«, and 36a2 6*c«x». 142 ELEMENTS OF ALGEBRA. Process. 24 a^b^c^ = 2^ X 3 X a^ X b^ X c^ ; 60 a^b^c^f = 2^ X 3 X 5 X a^ x 6^ x c^ X 2^2 ; 48 a^b^c^ = 2* X 3 X a^ X b^ X c^ ; 36 aH^c^x^ = 22 X 32 X a2 X 6^ x c^ X x^ .'. theH.C.F. = 22 X 3 X rt2 X 62 X c2=: 12a2Z>2c2. Explanation. Factoring each expression, it is seen that the only- factors common to each are 22, 3, a^, b\ and c^. Hence, all of these expressions can be divided by any of these factors, or by their product, and by no other expression. Example 2. Find the H. C. F. of 2 x^ - 2 a: 3/2, 4 x^ - Axy^, and 2 x* - 2 a:2 3/2 + 2 x3 3/ - 2 X ?/3 Process. 2 a:^ — 2 x 3/2 = 2 x (x + y) {x — y) \ 4 x^ - 4 xy* = 2^ X (x + y) {x - y) (a;2 + y^) ; 2 x^ -2x'^y^+ 2 x^ y -2xy^ = 2 x (x + y)^ (x - y); .-. the H. C. F. = 2 a: (a; + ?/) (a: - 2/) = 2 a; (a:2 _ y^). Explanation. Factoring each expression it is seen that the only factors common to each are 2, x, x + y, and x — y. Hence, all of these expressions can be divided by any of these factors, or by their product, and by no other expression. Note 2. If the expressions contain different powers of the same factor, the H. C. F. must contain the highest power of the factor which is common to all of the given expressions. Example 3. Find the H. C. F. of 8a^ x^ + IQa^x^ + 8 a^ x\ 2 a* a;2 - 4 a^ X - 6 a6, 6 (a^ + a xy, and 24 (a2 a; + a x^y. Process. 8a5x2+16a4x3+8a8x4= 23X a^Xx^ (a + xY; 2a^x^-4a^x-6a^= -2 X a'^X (a + a;) (3a-a;); 6(a2+aa:)2= 2 X3Xa2x (a + x^; 24(a2x + aa:2)8= 2^ X 3 X a^ X x^ (a + a:)3. The common factors are 2, a^, and a + x. .-. the H.C.F. = 2a2 (a + x). Hence, HIGHEST COMMON FACTOR. 143 To Find the H. G. F. of Two or more Expressions that can be Factored by Inspection. Separate the expressions into their factors. Take the product of the common factors, giving to each fac- tor the highest power which is common to all the given expressions. Exercise 56. FindtheH.C.F. of: 4. 12 a^lJ^x^ and 18 a^bs^ ; 6 ci^xy, 8 aT^y, dindi ^tii^xy^. 5. loa^a^y^, ^a^x^f, and 21d^x^7/. 6. 12 2:8^2 22^ 18 a:*/^^, and 36 a:^^^^. 7. 20 c8a:V, 8a2x2yl, and \2a^x^yi. 8. a^hx ■\- al?x and a^h — l/^. 9. a^y'^ — z^ and ax^y ^h xy -\- axz — hz. 10. 3a;*+8a:8+4ar*, ?^a^^-ll3^+^3^, and Za^l^7?-\2x^. 11. Za^x^y-Za^xy-Z^a^y and 3a2a:3-48 a'^ x - 3 a2 0^2 4. 48 a\ 12. x^-{-x,{x^ 1)2 and 2^8 + 1 ; a^" + a:" - 30 and a^- _ a:- _ 42; a:8 ^ 27, a:^ _ 9^ and 2 a^ + 5 a; - 3. 13. a^ — a^y, a^ — xy^, and 7^ — xi^. 14. a;* -f aj2 y2 _^ y4 ^nd a^ — 2ar^y+ 2xi^ — i^. 15. 12 (a - hf, 8 (a2 - ^2)2^ and 20 (a* - ?>4). 16. 8 a: 2; (a; — y) (x — z) and 12yz(i/ ^ x)(y — z). 17. 4ar»- 12a: + 9, 4a^- 9, and 4a^bx-6a^b. 18. a?-21f, x^-^xy^^f, and 2 r*- a;y~ 15/. 144 ELEMENTS OF ALGEBRA. 19. :r}' — if, (x^ — y^f, and ax^ — 1 axy + iS ay^. 20. ma^ — mx, 2x^-\-l^ x'^—2^ x, and 4:a?x^—4:a^x. 21. 24m7i-C?/i+16 ??,-4, 649i2-4, and IG^i^-S^i-f 1. 22. a;2 + 4 a; + 4, ^3 + g, and 4 ic2 + 2 a; - 12. 23. 16 ^3 _ 432, a;2 - 6 ^ + 9, and 5 x^ - 13 rr - 6. 24. rii^—n?, m7i — 7i^+mp—np, and m^—m?n + mn^—n^. 25. 6 x^ — 9Q X, m a^ y — 8 m y, and 15 ^ a:^ — 60 ^. 26. a;6«-ll2;3«+30, r^6«_i3^3«4.42^ a^i^j ^6n_^^3n_42. 27. a;3'' -125, ^2 « _ 10 ^« + 25, and 2 2^2« _ ^ ^n ^ 5^ 28. Sa^''- 125, 4^2«_25, and 4a:2«- 20^" + 25. 64. If the expressions cannot readily be factored by inspection, we adopt a method analogous to that used in arithmetic for the great- est common divisor of two or more numbers. The method depends on two principles : 1. A factor of any expression is a factor of any multiple of that expression. Thus, 4 is contained in 16, 4 times; it is evident that it is con- tained in 5 times 16, or 80, 5 times 4, or 20 times. In general, Since a factor is a divisor, if a represent a factor of any expression, m, so that a is contained in m, b times, it is evident that it is con- tained in r m, r times h, or r h times. 2. A common factor of any two expressions is a factor of their sum and their difference, and also the sum and the difference of any multiple of them. HIGHEST COMMON FACTOR. 145 Thus, 4 is contained in 36, 9 times, and in 16, 4 timed. Hence, it is contained in 36 + 16, 9 + 4, or 13 times, and in 36 - 16, 9— 4, or 5 times. Again, 4 is contained in 5 times 36, 5 times 9, or 45 times; also, 4 is contained in 10 times 16, 10 times 4, or 40 times. Hence, it is contained in 180 + 160, 45 + 40, or 85 times, and in 180-160, 45 - 40, or 5 times. In general, Let a be a factor of m and n, so that a is contained in m, b times, and in n, c times. Then (m + 7i) + a = 6 + c ; also, (m — n) -{- a = b - c. Again, since a is contained in m, b times, it is evident that it is containe<l in r m, r times b, or rb times ; also, since a is con- tained in n, c times, it is contained in s n, « times c, or sc times. Hence, rm -^ a=irb, and sn -r a = sc. Adding these equations, we have {rm -^ s n) ■Ta = rb-\-sc; subtracting the second equation from the first, we have (rm — .<* n) + a = rb — sc. The last two et^uations may be written (r 7n ± s n) -^ a = r b J^ s c. Therefore, rm±sn contains the factor a. , Example 1. Find the H.C. F. of 4 a:« - 3 a:^ - 24 a; - 9 and 8 x» - 2 or^ - 53 X - 39. Solution. The H.C. F. cannot be of higher degree than the first expression. If the first expression divides 8 x*—2 x^ — b'Sx — 39, it is the H.C. F. By trial, we find a remainder, 4 2:* — 5 a; - 21. The H.C. F. of the given expressions is also a divisor of 4 x* — 5 a: — 21, because 4x* — 5x — 21 is the difference between 8 a:* — 2 z^ — 53a: -39 and 2 times 4 a:» - 3 a;* - 24 z - 9 (Principle 2). Therefore, the H.C. F. cannot be of higher degree than 4x' — 5z — 21. If 4 z^ - 5 a: - 21 exactly divides 4 x« - 3 r* - 24 z - 9, it will be the H. C. F. By trial, we find a remainder, 2 z^ - 3 z - 9. The H.C. F. of 4 z2 - 5 z - 21 and 4 z« - 3 z* - 24 z - 9 is also a divi- sor of 2 z* — 3 z — 9, because 2 z* — 3 z — 9 is the difference between 4 z» - 3 z» - 24 z - 9 and z + 1 times 4 z^ - 5 z - 21 (Principle 2). Therefore, the H. C.F. cannot be of higher degree than 2z* — 3z — 9. If 2 z* — 3 z — 9 exactly divides 4 z'' — 5 z — 21, it will l)e the H. C. F. By trial, we find a remainder, z - 3. The H. C. F. of 2 z* — 3 z — 9 and 4 z* — 5 z — 21 is also a divisor of z — 3, because z — 3 is the difference between 4z* — 5z — 21 and 2 times 2z* — 3z — 9 (Principle 2). Therefore, the H.C. F. cannot be of higher de- gree than z — 3. If z — 3 exactly divides 2 x* — 3 z — 9. it will be 10 146 ELEMENTS OF ALGEBRA. the H.C.F. By trial, we find that a: — 3 Ib an exact divisor of 2x^-3x-9. Therefore a; - 3, or 3 - a; is the H. C. F. Process. 4x^-3x^- 24a: - 9 ) 8a;3 ~ 2a;2 - 53a; - 39 ( 2 2 times the divisor, 8 a:^ — 6 x^ — 48 a; — 1 8 First remainder. 4a,2- 5a: -21 4^2 _ 5 X times second divisor, Second remainder, X- -2l)4a:8-3a:2-24a:- 9(a:+l 4a:3-5a;2-2la: 2x2- 2x- 9 2 times third divisor. 2: r2-3a:-9)4x2-5a:-2l(2 4x2-6a:~18 Third remainder. X- 3 2 a: times fourth divisor, Fourth remainder, 3 times fourth divisor, x-3)2x2_3x-9(2x + 3 2x2 -6x 3x-9 3x-9 Therefore, the H. C. F. = x - 3, or 3 - x. Note 1. The signs of all the terms of the remainder may be changed : for if an expression A is divisible by — £, it is divisible by + B. Hence, in the above example, the H. C. F. is a: — 3, or 3 — a;. Example 2. Find the H.C. F. of 4x^ - x^y - xy^ - 6 y^ and 7x8 + 4x2^/ + 4X^2_3^8 Process. 4x8— x2i/—x2/2— 5 2/8)7x8+ 4^2?/+ Axy^— 33/8(7 4 times first dividend, 28x8+16x2 2/+16x?/2— 12^/8 7 times first divisor, 28x8— 7x^y— 7xy^—Sby^ First remainder, 23x2^+23xp+23/ = 23 ?/(x2+xi/+.y2) x2+xy+2/2)4x8— x^y— xy^—5y^(^4x- 6y Ax times second divisor, 4x8+4x23/+4x.?/''^ Second remainder, —bx'^y — bxy^—by^ — 5y times second divisor, —bx^y — bxy'^—b y^ Therefore, the H. C. F. = x2 + xi/ + ?/. Explanation. Arrange according to descending powers of x, take for the divisor the expression whose highest power has the smaller coeflficient, and multiply the dividend by 4 (to avoid fractions). Since 4 is not a factor of 4 x^ — x'^y — xy^ — 5 HIGHEST COMMON FACTOR. 147 given expressions is the H. C. F. of 4 x* — a;'* y - xy^ — b y* and 28 x« + 16 a^»y+ 16 a: y2- 12 i/« (Principle 2). Since 23 y {x^ -{- x y -\- y^) is the difference between 4 times the dividend und 7 times the divisor, the H. C. F. oi the given expressions is a divisor of it (Principle 2). Therefore, the H.C. F. cannot be of higher degree than 23y (x^ -^ ^y + y^) If the first remainder exactly divides the first divisor, it will be the H.C.F. Since 23 y is not a factor of the first divisor, it can be rejected. Therefore, x^ -\- xy + y^ is the H.C.F. This method is used only to determine the compound factor of the H. C. F. If the given expressions have simple factors, they must first be separated from them, and the H.C.F. of these must be reserved and multiplied into the compound factor obtained. Thus, Examples. Find the H.C.F. of 54x^y + 60x'^y^ - I8x»y* - 132 a:V and 18x«ya - 50 a^^y* + 2x*y*- I2x^y\ Solution. Removing the simple factors 6x^y and 2x^y^j and reserving their highest common factor, 2x*y, as forming a part of the H.C.F., we are to detennine the compound factor of 9a:* - 22x'^y^ -3xy*-\-l0y* = A and 9x* - 6x^y -\- x^y^ - 25 y* = B. UA exactly divides B, it is the H.C.F. of ^ and B. By trial, we find the remainder -y(6x*-23x^y- 3 x !/« + 35 .y«). The H. C. F. of A and B is also a divisor of this remainder, because the remainder is the difference between B and 1 times A (Principle 2). Reject -y from this remainder, since it is not a common factor of A and B, and represent the result by D. The H.C.F. of Z) and 2 A (a multiple ofi4) is the H.C.F. n\ A ami J5 (Principle 2). This cannot be of higher degree than D ; and if D exactly divides 2-4, it is the H.C F. By trial, we find a remainder, 153 y^ (3 x^ - x y - 5 y^). The H.C.F. of D and 2^ is also a divi.sor of this remainder. Reject 153 y«, and represent the result by E. The H.C. F. of E and D is the H.C.F. of D and 2 A ; and if E exactly divides D, it is the H.C. F. By trial, we find that E is an exact divisor of D. There- fore, E is the H.C. F. of A and B. Hence, the H.C. F. of the given expressions is 2x^y (3x^ - xy - 5 y^). 148 ELEMENTS OF ALGEBRA. Process. 9x^22x^y^-'3xy^+10y^ = A )9x*-Qx^y+ x^y'^ -25y^=B{l 1 times the first divisor, 9x* -22xY-3xy^+10y^ -6 a;8 y+2S x'Y+'^ x 2/3-35 ?/'» = -y{6x^-23x^y-Zxy^+36y^) {3x + 2Sy 6x^-23x'2y-3xy^+25y^ = D) ISx* - 44xY- ^^y^+ ^Oi/* = 2A 3x times second divisor, l Sx*-^9x^y- 9x'^y'^+l0bxy^ Second remainder, 69^:^- 3bxY-^^^xy^+ 20?/^ 2 times second remainder, 138x3^- 1{)xY-222xy^+ 40y* 23 y times second divisor, l 38x^y-529xY~ G9xy^+S05y^ Third remainder, 459xV-153x2/8-7652/4 = I53y^(3x^-xy-5y'^) 3x^-xy-5y^ = E)6xf^-23x^y- 3xy^ + 35y^ = D{2x-7 y 2x times third divisor, 6x^~ 2x^y — 10xy^ Fonrth remainder, -2lx^y+ 7xy^-\-35y^ — 72/ times third divisor, —21x'^y+ 7xy^-]-35y^ Therefore, K.C.F. = 2x^y {3x^-xy~5y'^). Example 4. Find the H. C. F. of 90 x^ y^ - 200 x^ y^ - 10 x^ y* and 144 x* 7/ - 64 a: i/* - 16 x^ / - 144 x^ y^. Removing the simple factors lOx^y^ and 16 xy, and reserving their highest common factor, 2xy, as forming part of the H. C. F., arranging according to descending powers of x, we have Process. 9x^—xy^—20y^ ) 9a;8— Qx^i/— xy'^— 4y^{^ 1 times the first divisor, 9^^ — xy^—20y^ First remainder, —9x^y +16^^ — —y(9x^—l6y^) 9x2-162/2)9x3- xy^-20y^{x x times second divisor, 9x^—l6xy^ Second remainder, 1 5 x ?/2 — 20 i/3 = 5 ^/^ (3 x — 4 1/) 3x-4?/)9x2-16y''=(3x + 4i/)(3x-4 2/)(3x + 4i/ 3x + 4y times third divisor, (3x + 4y)(3x — 4y) .•. the H. C.F. = 2xy (3x — 4y). Hence, in general, HIGHEST COMMON FACTOR. 149 To Find the H.C.F. of Two Polynomials that cannot readily be Factored by Inspection, if the given expressions have simple factors, remove them and iirraiige the resulting expressions acconling to powers of a common letter. Take that expression which ia of lower degree for the first divisor; or, if both are of the same degree, that whose first term has the smaller coefficient. If there is a re- mainder, divide the first divisor by it, and continue to divide the last divisor by the last remainder, until there is no remainder. The last divisor will be their highest common factor. The highest common factor of the simple factors multiplied by the last divisor will give the H.C.F. sought. " Notes : 2. If the first term of the dividend or of auy remainder is not ex- actly divisible by the first term of the divisor, that dividend or remainder must be multiplied by such an expression as will make the fii-st term thus divisible. 3. Observe that we may multiply or divide either of the polynomials, or any of the remainders which occur in the course of the work, by any factor which does not divide hoth of the polynomials, as such a factor can evidently form no part of the H. C. F. Exercise 57. Find the highest common factor of : 1. ar3 4- 2 ar2 - 13 a: + 10 and 2:3 _j. ,2 _ 10 a; + 8. 3. a^-x^-5x-Ssind 3^-4x^-11 x- 6. 4 x*-93^+29x^-39x+lS and 4 a^- 27x^+56 a: -33. 5. 2^ — 5 ax^ 4- 4 a^x and sd^ — ax^ + Za?x^ — 3 a^x, 6. 2f-l0xf-^%x^y and ^x'^-^xif-^-Za^f-^s^y. 7. 2a:S-lla;2_9 and 4a;«+ ll2:*4-81. 8. 18 ar^ + 3 a: - 6 and 18 a:^ _,. 95 ^ ^ 104 a; + 32. 150 ELEMENTS OF ALGEBRA. 9. 15 m^ n^ — 20 m^n? — 65 m^n — 30 m^ and 2amn^ -\- 20 am n^ — 16 a mn— 186 a m. 10. 36 m^ + 9 m3 - 27?/i* - 18 ?7i5 and 27^57^2 - 9 m%2 -18 m* 72.2 11. S x^ — 3 X y + xy^ — y^ and 4:X^y — 5 xy^ + y'^. 12. mnx^ — 82 m 7i a; — 3 //i ?^ and m^ ti^ ^5 _|_ 28 ^^5 ^2^2 - 9 m^n\ 13. a;3 - 4 ^2 + 2 ^ + 3 and 2 ^* - 9 ^3 ^ 12 ^2 _ 7 14. 16 x^-^^xy + 10 7/2 and 6 :?:* - 29 2;3 7/ + 43 i«2 ^2 -20^?/^. 15. 2m^n — 10m^n7j'^+ IS m^ny^+ 224:mny^+2d4:ny^ and 4 m* 71 — 20 m2 ?i 7/2 — 48 in ny^ + 112 7i 7/*. 16. 27?2.'*a;4"- 27?i"2;3" - 4 7??,**^2« + 4^";:c" and 6m"a?5n - 18 m^'x'^'' + 12 77l"^3« _|. 6 ^«^2n _ (3 ^n^n Query. How many factors in this result 1 65. To Prove the Method for Finding the H. C. F. of any Two Algebraic Expressions. Let A and B represent the ex- pressions, the degree of A being either equal to or higher than that of B. Di- vide A by B, and let the quotient be m and the remainder D ; divide B by D, and let the quotient be n and the re- mainder E ; divide D l)y E, and let the quotient be r and the remainder zero; that is, E is supposed to be exactly con- tained in D. We will first prove that -B is a common factor of A and B. From the nature of subtraction, the minuend is equal to the sub- trahend and remainder. Hence, A = mB -\- D, B zz n D + E, and Process. B)A (m mB D)B{n nD E)D(r rE HIGHEST COMMON FACTOR. 151 D = r E. Since the division has terminated, E is a divisor of D. E is also a divisor of n Z) (Principle 1) and of n D -{■ E^ or B (Principle 2). Hence, £ is a divisor oi mB (Principle 1), and of mB + D, or A (Principle 2). Therefore, £ is a common factor of A and B. We must now show that E is the highest common factor. Every divisor of A and B is also a divisor of m B (Principle 1), and of A —mBy or 2) (Principle 2). Therefore, every divisor of A and i5 is a divisor of nD (Principle 1), and of B — nD, or E (Principle 2). But no divisor of E can be of higher dej^ree than E itself. Therefore, E is the highest common factor of A and B. 66. Let i4, B, />, E, etc. represent any polynomials. Let m represent the H. C. F. of A and B, n the H. C. F. of m and Z), and p the H. C. F. of n and E^ etc. Evidently m is the product of all the factors common to A and B ; also, n is the product of all the factors common to m and Z), and /) is the product of all the factors common to n and £, or /> is the product of all the factors common to A, By D, and £, etc., which is their H.C.F. Hence, in general, To Find the H. C. F. of Several Polynomials. Find the H. C. F. of two of them; then of this result and one of the remaining polynomials; and so on. The last result found ^vill be the H.C.F. of the given polynomials. Exercise 58. Find theH.C.F. of : 4- 4 y*, and 2 a;^ + 2 i/. 2. ai^ + a^-Sa^-^Jx-9, 2-^ x -\- 2^ -{- s^-\- 2x^ + 2x^, and 3 + 3a:2_^^4.^^4^ 3. m" a:3 4. 2 m" ar2 + m" J? + 2 m", 2 x'^ + 6 2^ + 4: s^, Za^-h93^+9x-\-6, '6x^- l23^-Sx^-6x, and Sx^ -\-2-\-5x-\-S2^. 152 ELEMENTS OF ALGEBRA. 4. 2x^-5 x+ 6- 3x\ 3 2;2 + 2 a;3 - 8 :i: - 12, and 5. Sa^"" - 33 a;2" + 96 a;" - 84, 68 x^" - 92 x^"" - 24a;» + 32 x!^\ a;3" + 11 ^"-6-6 :>j2«, 50 a;" + 20 r?;3« - 60 r2;2H - 20, 5 x^" - 10 a^" + 7 a;" - 14, and 3 ^«" - 35 a:^" + 162 2:^" - 372 a;3« + 494 a;2» - 192 x\ 6. 9 a:2« + 4 X" + 2 2:3" _ 15^ 48 ^n + 30 _ 343 ^" - 24 x^"", 8 a;2« + 4 x^'' + 3 a;" + 20, and 2 x^"" + 12 a:^" - 94 a;"- 60. 7. o 3^ — 2 X y^ — 5 x^ I/, 5 xy^— 6 y^ — 3 x^y"^ — x^y + x\ 9 a^ - 8 x^ y - 2i) X7/, S xy^ - 7 x^y - 2 y^ + 3x^ 10 y^ — x^y^ — 5 x^y + S x!^ — 7 xy^, and x^ — x^y — x^y^ - x]^— 2y^. Miscellaneous Exercise 59. Note. When possible the student should separate the given expressions into their factors by inspection. rindtheH.C.r. of: 1. 7^ — xy^ and x^ + x'^y ■\- xy + y\ 2. x^ — ?/2, {x — ?/)2, of — x'^y, and 2x'^ — 2xy. 3. 2 x^ — X — 1, X y — y, x'^ y — X y, and 3 x^ — x — 2. 4. rc6 - 6 X + 5, 2a^+ b-8x + x\ x^ + x^ - 11 x + 9, and 42 2^2 + 30 - 72 x. 5. a;2_i8a;+45, 22:2-7a:+3, 2^2-9, and 33p-7x-6. 6. 6a;4"- 3:r3n_^2n_^n_;^ ^^^ 3a^^ - 3a^'' - 2x^'' - cr" - 1. HIGHEST COMMON FACTOR. 153 7. a^ - //3^ x^ + ^V 4- 2/^ a:^ - //, x^ + x// + i/, a^ + sH^y — x-^ — i/y and r^// 4- ar^y^ + .^3^. 8. 2ar»+2rt + 4aa;, x^+23^+2x+l, 7b+Ubx + 7 6^-3+ 14 6r2, 3 :r2 - (3 m + n - 3) a: - 3 m - n, a^-_ 2 2^2 4- 1, and 2 2:2+ (2;? + ^ + 2) a: 4- 2p + q. 9. a:* - 27 &8 ic, (2:2 _ 3 5 ^)2^ a a^ - a b x^ - 6 a b^ x, and &2:*-4 62a:3 4. 353^, 10. 4 2:4'»-2 2:3«_^ 3a.»_9 ^nd 2 a:*" + 2:2'^ - 2 2:3» + 3 2:* - 6. 11. (a + 6) (a - ^/), (« + 5) (6 - a), and (ft + «)^(^^ - ^)^. 12. 2 63 _ 10 a &2 _^ 8 a2 6, 4 a2 _ 5 « J 4. ^,2^ ^4 _ ^,4^ 9a*-3a68+3a252_9a8j^ and 3 a^- 3 a26 + aJ2_ j3 13. 3 2:3'*-3m2.'2'» + 2m2a:''-2m8 and 3a:3"+ 2m2a:* + 8 m3+ 12 ma:3« 14. (ni — n) {x — y), (m ^n)(j/ — x), and (n — m) {x—y). 15. 9 2r»+ 3 2:3+ 12a:+ 20 + 2:^ 3 2:2^2 + ^^ _^2 2r» + 12y2+4y + 8, G2:2^_a;6_|.62^^32,4.24anda^»2^ + 3 2:2y + 4 2:2 + 4 y2 + 12 y + 16. 16. a3-+3a2'»6« + 3a'»62m_^^«^ Sa^^+oJ^'", 4a2"fe2'» + 12 a"fe3m + 8 ft*"", and a2-- 62«. 17. 2;* — ?n. 2:3 _|_ (^ _ 1) a;2 4. ^ 3. _ ^ ^j^^-l a:* __ ^ ^ + (w — 1) 2:2 ^ ,j 2; _ ^ 18. 3n2ar»+ 12?;i27i2+ 3n3^- 15mn^x+ \2m^nx — Ibmnx^ and 2 m 71 2:3 4. g ^^3 1^2 _ 2 n^ a:3 4. g ^3 ^ 2; + 2 m n^ x^ — ^ m? n^ X — 2 noi^ — ^ m^ nx^. 154 ELEMENTS OF ALGEBRA. 19. x'^ — ma^ — mi? x^ — iii^ x — 2 m^, a;^ — 6 ni^ + mx, x"^ — 2 m^ — m x, 3 3^ — 7 m x"^ + S m^ x — 2 m^, and x^ — 8 7n'^ + 2 7?i X. 20. 12x^i/-24.3^2/+ 12x^7/, {x^y-xf'f, xy{x^-ff, and ^7^y^-2^x^if^ 2^x^i/-^fx. 21. a^- 2o?h - aV^+ 2h^, a^ + a^h - ah^ - h\ «3 _ 3 a 62 + 2 63, a^ -h^, 2>ac-3hc + 2ah- 2V\ a^-b\ and 2h^ + a c - he - 2 ab. 22. a2 _ (^ ^_ c)2^ (^ 4. ^)2 _ j2^ c2 - {a + hf, and a^ + 2a6 + 62+ 2 6c + 6-2 + 2ac. 23. a^e^ + a^s^-Se, ^4'^+ 5 2:3«_|_ g ^n^ :i:6«_4 2,3"_96^ ^3«_|. 32:2« + 3a:"+2, a:^'^ - 9 2:2" + 20, ^nd 3a:3n^3^2. + 5 2f^ + 2. 24. ^ - 2 2^2 + 3 ^ _ 6 an(i ;z;4 - a?3 - ^ - 2 oj. 25. 4 ^ 2/^ ~ 2 2/3 + 6 ^2 ^ and 4 ^2 ^ ^ ^ x^ — A.xy^. 26. 35^4-47^2_|_i3^+X and 42^4+41;2^3_9^2_9^_l 27. m7z,3+2??z7i2+7'/i'/i+2m and 3?i^— 12 ?^3_ 37^2^5^^ 28. 2m22/5+166m22/2-96m2 2/ + 108m2 and ^mTv^f — 144 m 1^ y^—l^m 7? ?/2 — 108 m n^. 29. 2^4_6^.3^3^2_3^,+l and ^7_3^6+^_4^2+i2;r-4. 30. 4a;H322;3+36^2^8^ and 8^6_24^4+24a;2-8. 31. a;^"— 82/3"»aj2«__2;"2/'» + 2/'" and ^2«__4aj«^+42/2m LOWEST COMMON MULTIPLE. 155 CHAPTER XIII. LOWEST COMMON MULTIPLE. 67. A Multiple of a uumber coutains all the factors of the j^aven number with higJiest powers. Thus, since 24 = 2» X 3, 2« X 3 is a multiple of 24. A Common Multiple of two or more numbers contains all the factors of the given numbers with highest powers. Thus, since 12 = 2^ X 3 and 9 = 3^^, 2^ X 3^ is a common multi- ple of 12 and 0. The Lowest Common Multiple (L. C. M.) of two or more algebraic expressions is the expression of lowest degree which can be exactly divided by each of them. Thus, 6 a*x»y« is the L. C. M. of 6 a*, x y\ a:», and a«y«. Example 1. Find the L. C. M. of 42 a^x y*, 56 a x*y^, 63 a«x*i/«, and 21 a* x^y. BolutioiL Separating the expressions into their factors, we have 42 a»a? y* = 2 X 3 X 7 X a*^ X x X y*, 56 a x*y^ = 2* X 7 X a X x* X y^, 63 a»x«y« = 3« X 7 X a» X x« X y», 21 a*x»y = 3 X 7 X a* X x« X y. 2* X 3* X 7 is the least common multiple of the coefficierts 42, 66, 63, and 21 ; a* is the lowest power of a that can be evenly divided by ich of the factors a*, a, a\ a* ; x* is the lowest power of x that can Ihj evenly divided by each of the factors x, x*, x^, x^ ; y* is the lowest power of y that can be evenly divided by each of the factors y*, y^, y*, y. Hence, the L. C. M. = 2» X 3^ X 7 X a» X x« X y» = 504a«x*y«, 156 ELEMENTS OF ALGEBRA. Example 2. Find the L.O.'M.. oi 6x^-2 x, 9x^-3 x, t5{z^+x y), 8 (x z/ + y'^y\ and \2 a^ x^ y^. Solution. Separating the given expressions into their factors, we have 12 a^-x^y' = 22 X 3 X a2 X a:3 X y^ 8(^2^ + 3/2)2^23 X 2/2 X (a: + y)2, 6(a;2 + ar2/) = 2 X 3 X a; x (a; + 2/), 6a;2-2a;=2 x a; X (3 a;- 1), 9a;2-3a:= 3 X a; X (3 a;- 1). 23 X 3 is the least common multiple of the coefficients ; aP- is the lowest power of a that can be evenly divided by a^ ; ofi is the lowest power of X that can be evenly divided by each of the factors x\ x, x, x. Similarly 2/, (a? + 2/), and (3 a: - 1), each affected with its highest ex- ponent, must be used as multipliers. Therefore, the L. C. M. = 2^ X 3 X a2 x a;^ x 2/^ X (x+yf X (3 a;- 1) = 24 a^x^y^ (x + y)" (3 x - 1). Example 3. Find the L.C.M. of 4 aa;2 2/2 + n aa;2/2 - 3a t/^ a;8 + 6 a;2 + 9 a;, 3 x^ y^ + 7x'^y^-6xy% and 24 aa;2- 22aa:+ 4 a. Process. 4aa;22/2+ llaa;2/2-3a2/2= a X y^(x + 3) X i^x-1), a;3 + 6a;2 + 9 a; = xX (x + 3)2, 3a:32/3 + 7a;22/3-6a;.?/3= x y^(x + 3)X (3a;-2), 24aa;2- 22aa; + 4a = 2a X (4a;-l) (3a;-2). .-. the L.C.M. = 2aa;iy3(a; + 3)2 (4 a;- 1) (3 a;- 2). Hence, in general, To Find the L. C. M. of Two or more Expressions that can be Factored by Inspection. Separate the expressions into their factors. Take the product of the factors affecting each with its highest exponent. Note. The L. C. M. of two or more prime expressions is their product. Thus, the L.C.M. of (i^ + ab + b^, aS + b% and a^ + b^ is (a^ + ab + b^) (a^ + 62) (^s + J8). LOWEST COMMON MULTIPLE. 157 Exercise 60. Find the L CM. of: 1. 4&3^i/, ^^aT^f, and 63r/^2«. 2. 24:111 71^3^, Z(Sm^n^2^, and 4871828. 3. Ua^l^c^, 9aHc2, and ^(Sah^d^. 4. 12m*n2 2/3, l^mnf, and 24m^?i3. 5. 12aa;8y*, a:'"^ — ?/-, 2;-— 2a:y + ?/, and rr^ + 2a;^ ^- 2/2. 6. m^ (a;^ — ^), 71^ (^x — y), and a:"* — y^. 7. 2a:(a;- y), -ixyix^ --?/), and 62:3/2(2.4. y)^ 8. 2:2 + a: - 20, 2^^ - 10 a: + 24, and 2:2 _ 2: _ 30. 9. 2^2+22:, 2:2 + 4a: + 4, 2:243^42, and a^ -\- o x -^^ e,. 10. 2:* + a2 2:2 + a* and 2-'* — a a:^ — ft^ a; + a*. 11. a:2 _ 3 2; - 28, 2^2 + 2: - 12, and a^ - 10 a: + 21. 12. 15 (2:2^ -xy% 21(2^- ar/), and 35 {xy"^ + f). 13. ar2 - 1, ar^ + 1, and a^ - i. 14. '6x^-\- llx-^ 0, 32:2 4. 3^, _j. 4^ g^^j 2r^ + 52- + 6. 15. 2^+{a-^b)x-\-aby a^-\-(a-\-c)x+ac, and a^-{-(b-\-c)x-{-bc. 16. mx — my — nx + ny, (x — y)^, and Zm^n—?nnn^. 17. a:2 4. (rt 4 2,) 2, 4 a 5 and ar2 + (a - i) a: - a 6. 18. ar2 - 1, a^2 4 1, 2:4 4 1 ^nd a:^ + 1. 19. a:^ + ar*y + a;?/2 -f ?/3^ ^i — x'^y -{- xy^ — if, and ^ -\- x^y -xf-f. 158 ELEMENTS OF ALGEBRA. 20. 6 aa^+7 a^x^-S a^x, 3 a^ x^ + Ua^x-^ a\ and 6x^ + 39ax + 45 a^. 21. x^ + 5 X + 4:, x^ + 2 X - 8, and a:^ + 7 ic + 12. 22.* 12 x^ - 23 a:^ + 10 2/^, 4a:2 _ 9^^ _j_ 5^2^ ^nd 30^2 -— 5xy + 2y^. 23. a'-^-4&2, a3-2a2Z^+4a&2_8j3^ and a^+2a^b + 4 a 62 + 8 53 24. a m + <z '/I + & /?i + & rt and ax + ay + hx-^hy. 25. 8a;2_38a;2/+352/^ 4a:2_:i,z/_53/2 and 2x^-5xy-7y^ 26. " 2^2 + ?/2, x^ — n:2 2/2 + 2/^ and 2:^ + y^. 27. 60 a;4 + 5 a;3 _ 5 ^2^ 60 n;2 ?/ + 32 a; 2/ + 4 y, and 40 a:^ 2/ ~ 2 a:2 2/ — 2 a;?/. 28. {a + 6)2 - (c + ^)2, (a + c)2 - (& + ^)2, and (a + df - {b •+ c)2. 29. 2^2 -{_ ^ ^ _|_ ^2^ 2^ _ ^ y _l_ y2^ and a:^ + a;2?/2 + 2/*. 30. 3 a^* + 26 a;3 + 35 a;2^ 6 a:2 + 38 a: - 28, and 27 x^ + 21x^-30x. 31. 12 a:2n _^ 3 ^n _ ^g, -^^ ar^« + 30 a;2" + 12 x\ and 32^2n_4Q^„_ 28 32. « (??/ — n), b (n — m), and — c{m — n). 33. (a -b)(b- c), (b -a)(b- c), and (b -a)(c- b). 34. a(& — a:)(a;— ^), b{c—x){x—a), and c(a — a:) (a:— 6). 35. a:*^ - 2 aj2n _,. 1 ^nd a:*" + 4 x^'' + 6 a^" + 4 a:" + 1. Result. a;«" + 2 ar^" - a;^" - 4 rr^" - x^" + 2 a;" + 1. LOWEST COMMON MULTIPLE. 159 68. If the expressious cannot be factored by inspection, find their H.C. F., then proceed as before. Thus, Example L Find the L. C. M. of 2 z< + a:« - 20 x" - 7 x + 24 and 2 X* + 3 z» - 13 x2 - 7 X -h 15. Solution. The H.C.F. of the expressions (Art. 64) is x^-\-2x-3. Dividing each expression (for the other factor) by x* + 2 x — 3, we have 2 X* - 3 X — 8 and 2 x* — x - 5. Hence, 2 x« + x» - 20 x2 - 7 X -h 24 = (x2 -h 2 X - 3) (2 x« - 3 X - 8), 2 X* + 3 x« - 13 x2 - 7 X -h 15 = (x* -f 2 X - 3) (2 x'^ - x - 5). .-. the L.C. M. = (x2 -H 2 X - 3) (2 x2 - 3x - 8) (2 x2 - X - 5). Example 2. Find the L.C.M. of x« - Sx^ -f- 19z -h 12, x»- 6x2 + 11 X - 6, and x« - 9 x2 -f 26 X - 24. Solution. The H.C.F. of the expressions (Art. 66) is x - 3. Dividing each of the expressions by x — 3, and factoring the quotients, we have x»-8x«+19x-12 = (x-3)(x2-5x + 4) = (x -3) (x-1) (x-4), x»-6xHnx- 6= (x-3)(x2-3xf2) = (x-3) (x-1) (x-2), x«-9x2-|-26x-24= (x-3)(x2-6x-f8) = (x-3) (x-2)(x-4). Therefore, the L.C.M. = (x-3)(x-l)(x-2)(x-4) = x*-10x»+35x2-50x + 24. Hence, To Find the L C M. of Two or more Polynomials that can- not readily be Factored by Inspection. Find the H.C.F. of the ^^iven polynomial!*, and divide each polynomial by it. Then find the L. C. M. of their quotients, and multiply it by the H. C. F. Exercise 61. Find the L. CM. of: 3. x^-{-2.x-3, :r?-\-Sx^-r:-S, and a:^ + 4 r^ + a: - 6. 160 ELEMENTS OF ALGEBRA. 4. ic* — m aj3 — ni^ x^ — m^ x — 2m^ and 3 a;^ — 7 m cc^ + 3 m^ 5:: — 2 m^. + 38^2/3+ 16^2/*- 10 2/^- 6. a?3 - 9 :c2 + 26 ^ - 24, a;3 - 10 a:2 + 31 2; - 30, aud aj3- 112^2+ 38:^-40. 7. ^-4 -2)3- 4 a;2+ 16 2^-24, 2^3_ 5^2 ^_ 8^-4, and a:2 + 2 2: - 8. 8. 2)3 _^ ^.2 _ 10 a: 4- 8, ic2 + 2 a: - 8, a;2 - 3 a; + 2, and ic2- 1. 9. 6 2;3 + 15 0^2 _ 6 ^ 4. 9 and 9 a;3 4. G 2,2_ 51 ^ 4. 36^ 10. 2 ar^ - 8 a;4 + 12 a;3 - 8 2:2 + 2 a:, 3 a;5 - 6 2^ + 3 re, aud a::3_3^2_|_3^_l 11. a:* + 5 2^3 + 5 rc2 - 5 2: - 6, a:^ + 6 a:2 + 11 a; + 6, and a:^ + 4 2^2 + 2^ — 6. 12. 2 2)3 + 7 a;2 + 8 X + 3, 2 x^ - 2^2 - 4 2; + 3, 2 a:^ + 3 2:4 + 2 x^ + 3 ^,2 4_ 2 a: + 3, and a?* 4- 2^2 + 1. 69. To Prove the Method for finding the L.C.M. of any Two or more Algebraic Expressions. Let A, B, D, E, etc. represent the expressions, F represent their H. C. F., and M represent their L. C. M. Also, let a, b, d, e, etc. represent the respective quo- tients when Af B, D, etc. are divided by F. Then, A =aF, B = bF, D ^ d F, E = e F, etc. (1) F is the product of all the factors common to A, B, Z>, etc. The quotients a, b, d, e, etc. have no common factor. Hence, their L.C. M. is a 6 rf . . . , etc. and the L. C. M. of aF, bF,dF, etc., or their equals A, B, D, etc., is ah d ... F. Therefore, M — abde F, etc. LOWEST COMMON MULTIPLE. 161 70. Let A, B, D, Ej etc. represent any polynomials. Let A' represent the L.C. M. of A and B, P the L.C. M. of N and D, and R the L. C. M. of P and Ej etc. Evidently R is the expression of lowest degree which can be divided by P and E exactly ; also, P is the expression of lowest degree which is exactly divisible by A^ and Z>, and N is the expression of lowest degree which is exactly divisible by A and B. Therefore, R is the expression of lowest degree which is exactly divisible by ^4, ^, 2>, and E, etc Hence, To Find the L.G.M. of Several Polynomials. Find the L. C. M. of two of theni; then of this result and one of the remaining expressions; and so on. 71. Let A and B represent any two expressions. Let F repre- sent their H.C. F., and M represent their L.C. M. Also, let a and b be the respective quotients when A and B are divided by F. Then A = aF, B = bF, and M = ab F. Multiplying the first equations together (Axiom 3, Art. 47), we have AxB = aFXbF=FXabF. Therefore, substituting for abF its value M, A B - F M. Hence, in general, The Product of any Two Expressions is Equal to the Product of their EOF. and L.C.M. Miscellaneous Exercise 62. Find the L. CM. of: aZ^a^h-ah^-h^ and a^ - 2 aH - a 6^ + 2h\ 2. a:4« _ 10 2:2- 4. 9^ ^n 4. 10 r^" 4- 20 ar** - 10 a:"- 21, and a;*- + 4 a;8" - 22 ar»" - 4 af 4- 21. 3. 2:3"-4ar2"3r+ Oa:"^^"'- lOyS" and a:«*-f2a;2nym 11 162 ELEMENTS OF ALGEBRA. 4. s^ -^ Sx'^ + x^ + 3x^ + x + S, 2a^+6x^^2x-6, r" + 2x^ + a^+ 2x^ + x+2, and 2a^+3x^+2a^ + 3x'^ + 2;:c+ 3. 5. X y — b X, X y — a y, i/^ ^ 3 h y -\- 2 h^, x y ~ 2 h^, xy — 2hx — ay + 2 ah, and xy — hx — ay -\- ah. 6. a^" + ^^4" h"^ + a3» 52m _j_ ^2n 53m _^ ^n 54m ^ j5m^ ^nd 7. .:c2'* - 4 a'-^"*, .:c3'' + 2 a"' *2« + 4 a^'"^;" + 8 a^^ and ^3« _ 2 ^'« ^2» ^_ 4 ^2m ^« _ 3 ^3m 8. 27^"" + {2a-3 &)^2n _ (2 52 + 3a&)a;" + 3 ^)3 and 2a;2"- (3 6-2 c)^?'^- 3 6c. 9. ^- 2:r2_^ 4^_8^ ^3+2^2_4^_8^ a^-3a^ -4:X + 12, and ^^ - 3 a;* - 20 a;^ + 60 aj2 + 64 a; - 192. 10. x^''-{a-h)x^-ah, x'^''-(h-c)x''-hc, a^^'-x^^'h^ - a;6»62 + 58^ and a)2» - (c - a)x'' - ac. Find the H. C. F. and L. C. M. of : 11. 3 o[^ — 7 x^y + 5xy^ — y^, x^ y + 3 xy^ — 3 0^—1^, and 3 a^ + 5 x^y + xy^ — y^. 12. 6^'5 4.i52,4^_4^2^_;l^Q^2^2^y and9^y-27a^22^ — 6 a2a;?/ + 18 <x^z/. 13. 6^3« + ^2n_5^n_2 and 6 2^" + 5 2;2~-3.2J*-2. 14. a^ -ah + h^ a^ + ah + h\ a^ + a^ h'^ + 6^ a^ + j3^ a^ — 6^, and (c^ — h^f. 15. 2rc2^(6^_io6)a3-30a6 and 3 a;2- (9 a + 15 6) a; + 45 a 6. LOWEST COMMON MULTIPLE. 163 16. a:3n_ 92«J- ^ 26 X-'* - 24 and a:3»_ 122:2-+ 47 ^•"-CO. 17. (a.-2 + 62)c + (62 + c2)x and {x^ - W) c + {h^ - c^) x. 18. (2 2^2 _ 3 ,,i2j y 4. (2 m2 - 3 y2) a; and (2 m2 4. 3^2)^ + (2x2+3m2)y. 19. 2:3«^ 2a'"a:2n_^ a2'"a:" + 2 a^^ and a:^" _ 2 a'"aj2« + a2m^„_2a3'". 20. a;3 +3a^y + 3a;^^/J + 2/, oi^ - x^y ^ xy^ - 2f, sc^ -\- xy -\- y^y and «* + 2^2 ^2 _^ ^ 21. 20^^x^-h 25a:< + 5a:8~2:~l, and 25 a;*-10 0:24.1, 22. 2:8* — yS* a^s-y- — i/*", y^(af — f)^, and a:2« - 2 a;*y" + 2^'". Find the L.C.M. of: 23. a:*-7a:8_7-2^_^43a.4.42 and ar*-9a:8+9a:3 H- 41 a; - 42. 24. ir3+4a:a+6a;+9, «:«+«»- 2 a; +12, and a:2_^_i2. 25. 4a:«-42:*~29a^»-21 and 4a:«+ 24a;* + 41 2:24.21. 26. 2a:*-ll2:8+32r24.i0a; and 32:*-142:«-6a;24.5a., 27. 2:3 _ 6 2-2 4. 11 a; _ e, a;3 _ a:2 _ 14 a; + 24, and 2:3 + .r2 - 17 2: + 15, 28. 32:^ + 52:84.525^52.^2 and 32^-2:8+2:2_2._2. 29. 9a:*+18a:3_.^_92^4.4 and 6a:*+17a:«-10a:+8. 30. 2 m^ + m2 - m + 3 and 2 m8 + 5 m2 ~ w - 6. 164 ELEMENTS OF ALGEBRA. CHAPTER XIV. ALGEBRAIC FRACTIONS. 72. The expression (a + &) -f (m + n) may be written — 77- . It is read the same in each case. The second form is called a Fraction ; the dividend is the Numerator, the divisor is the Denominator, and the two taken together are called the Terms of the fraction, a, b, m, and n may- represent any numbers whatever. Hence, An Algebraic Fraction is an indicated operation in divi- sion. A Mixed Expression is one composed of entire and frac- tional parts ; as, n m-\ a Note. The dividing line has the force of a symbol of aggregation, and the sign before it is the sign of the fraction and belongs to its algebraic value. 73. Multiplying or dividing the divisor and the dividend by the same number does not change the quotient. For, if we multiply the dividend by any number, as m, the quotient will be increased m times ; if we multiply the divisor by m, the quotient will be dimin- ished as many times. A similar method of reasoning may be applied to the dividend and divisor. A fraction is in its lowest terms when the numerator and denominator have no common factor. 7a^hc Example 1. Reduce ^^^ 3,3 to its lowest terms. ^o 0/ C ALGEBRAIC FRACTIONS. 165 Solution. The H.C.F. of the numerator aud the denominator 7a^bc X 1 is la^bc. Factoring, we have 7^2^^ X4ac ' ^^J^^^"^8 ^^^ H.C. F., we have Since the terms are prime to each other the fraction 4ac is in its lowest form. 6 a* + ax- 15 x^ Example 2. Reduce ,r: 2 . ^u^^ — TTZz to its lowest terms. 15 a* + lis ax — 15 x^ 6a^-\- ax - 15 x' _ (3 a + 5 x) (2 a - 3 a;) ^^°*^®**' 15a2+ 16aa:-15x2~ (3a4-5a;)(5a-3x) _ 2a -Sx - 5a-3x' Explanation. Dividing the terms of the fraction by their H. C. F., we have This result is in its lowest terms, since the 5a — 3x numerator and denominator have no common factor. x*-j-x^y-\-xy* — y* Example 3. Reduce -7 — ^ 5 -. to its lowest terms. XT — x*y — xy* — y* x^ + x^y+xy'-y* _ (x* - y*) + (x*y f x y*) Process. ^^^y_^^_yA - (^ _ y4) _ (^^ ^ ^^.^ ^ (x ^ + y«) (x« -y^) + xy (x« + y«) l^ + y«) (a:« - ya) - xy (x« + y^^ ^ (x« + y«)[x«-y« + xy] (x2 + y«)[x2-y2_^yj x^ + xy-y^ When the factors of the numerator and denominator cannot be readily found by inspection, their H. C. F. may be found by the method of Art. 64, and the fraction then reduced to its lowest terms. Thus, . T> , 4a»+ 12a«6 -afta- 15!»» . , Example 4. Reduce ^ . , ..^ «. -. — i» — Trut to its lowest 6 a" + 13 a*o — 4 ao* — 15 6* terras. 166 ELEMENTS OF ALGEBRA. Solution. 6a^+r3a^b-4ab^- 1568) 4a^+l2aH~ ab^-l5b^(^2 3 times the numerator, 1 2 a^ +36 a^lSab^ —45 b^ 2 times the denominator, I2a^-j-26a^b-8ab^-30b^ First remainder, lOa^b + bab^—lbb^ = 5b(2a^+ab-Sb^). 2a2 + a&-362)6a8+ 13a26-4a62_ i568(3a + 56 6as+ 3a2&-9a62 10 a^b + 5 ab-^- 15 b^ 10 a^b-{~ 5 ab^- 15 68 .'. the H. C.F. of the numerator and denominator is 2a2+a6— 36^. Dividing each term of the fraction by 2 a^ + a 6 — 3 6^^ we have 4a8+ 12a26-a62_i5 58 2a + 56 6a8+ 13a26-4a62- 15 68 3a + 56 Hence, To Eeduce a Fraction to its Lowest Terms. Divide both terms by their H. C. F. Exercise 63. Keduce to lowest terms : 75ax^y^' ^m^a^y'^^ 4:X^ + 6 xy' 72 m^n^x^\ mn^{a^-y^f ^ 2^ + Sx+ 1 24:'m'nix''' w?n{a^ — j^)' x^ — x — 2 6 m^- 11m -10 20 (a^ - if) af"!/^" 6 w2- 19 m +10' 5 2:2 +5 ^2/ + 5^/2' ^imyn+i- 3 m^ + 23 yyi - 36 3 772^ + ^m^n + ^Trv^Ti^ 4 m2 + 33 m - 27 ' m* + m^Ti - 2 m27i2 * ^ in + Zmx x^ — {a -\- b)x + ab 4 ml — 4 mt 2:2 ' a:2 + (c — ^t) 2: — « c ' (m + w)2 — a:2 ^.s _ 3 ^2y + 3 2; y^ _ -^yS m a: + 71 a: — a:2' a:^ — x^y — xy^ -\- y'^ ALGEBRAIC FRACTIONS. 167 cr^ — (y + mf ac — ad — he -\- hd x^ -\- X y ■{- m X* a^ — b^ 0.^4- (g + b)x+ ah 27a + a^ a^Jt\a-\-c)x+ ac' 18a-6a2+2a8' 10. 11. 12. (a _ 6)2 _ c2' (6 4- a;)-^ - {a + c)2 m^ — m* n — m n* + 71^ a a^ — 6 ic*"*"^ m* — m^ 71 — m^ ?i2 -|- ??i ti^' a^hx — h^a^ a8+3aH+ 3a62+268' 48 ar* + 16 a:2 _ 15 ' 7*. Example. Reduce :7^ :r— ^ — to a mixed expression. ProcesB. 2x - 3y ) 4x« - 16x2y + 29a;i/2 - 22y« ( 2z2 - sxy -f 7ya + 5-^^^^4— -10x2y + 29x1/2 -10xgy-H5xy« 14xy^-22y« 14xy«-21y« - .y» • Explanation. Dividing the numerator by the denominator, we have 2 «* — 6 xy + 7y'' for a quotient, and a remainder of — y«. As — y* is not exactly divisible by 2x - 3y, we indicate the division and atld the result to 2 x* — 5 x y + 7 y*. Hence, To Reduce a Fraction to an Entire or Mixed Expression. Divide the numerator by the denominator. 168 ELEMENTS OF ALGEBRA. Exercise 64. Eeduce to an entire or mixed expression 1 + 22J 2x^-x^- 9x^+14: ' r^^^J^' 2x^- x-S x^-2x^ 6 a3 - 13 o2 + 6 a - 6 z. - 3. 5. -x+ 1' 3a^-2a + 1 x^ + ax^-3 a^x- S a^ ^^3 4. 2 a:^ - 12 a: - 13 x-2a ' a;2 + ^ - 12 x^ — 2 x^y^ + f/ x^-\-(m + n+l)x + mn+a x^ + 2 xy + y^ ' x + n 6a^-5x^+ Ix-b a^S" - a:^"?/" + x^'y'^'' - y^ ^ 2x+l ' ^'" - 2/" 75. Every expression may be considered as a fraction whose denominator is unity. Thus, a = - ; a^b — c^ = . 2;2 _ y2 _ 5 Example. Reduce x -\- y to fractional form. ^ x-y a:2 - «2 _ 5 X + y x^ - y^ — t) Process. x-\-y — ~ ^ ^ {X- ^V) X - X (rc- -y x' ar2- 1 X(x- -5) 5 X —y ~ X — y' Explanation. Writing the entire part in the form of a fraction whose denominator is 1, and multiplying both terms of it by x — y, we have the third expression. Since the sum or difference of the quotients of two or more expressions divided by a common divisor, is the same as the quotient of the sum or difference of the expressions divided by the same divisor, we have the fourth expression. Uniting like terms, we have the result. Hence, ALGEBRAIC FRACTIONS. 169 To Reduce a Mixed Expression to the Form of a Fraction. Multiply the entire puit liy the denouiinator ; to the product annex the numerator; unite like terms and under the result write the denominator. Notes : 1. In the above example, since the sign before the dividing line indicates subtraction, we most subtract the numerator, x^ — y^ — 5, from {X + y) (x - y). 2. If the sign of the fraction is — , and the numerator is a polynomial, it will be found convenient to enclose it in a symbol of aggregation before annex- ing it to the product. Exercise 65. Reduce to fractional forms : 1. a — X -\ ; — ; - -\- I] a + r- . a -\- X m — n a — b 2. „,._,„, + ,.-_^^; ^^j;±_^^ -(.-,). , « — m^n^ a . „ . . i m* — 1 3. m n H ; m^ + m^ + ?>i -f 1 m n m — 1 4. ic + 1 4- ; ; -3-^ — 3 + 1 ; w (a: + ?/) + — — 7/r — ar x -r t 2 n (3 w2 + ?i2) wi* + w* 6. m + w ^ — , — ^2 — ; (w + ny — r^ • 7 a^m _a^,^«^ y8»_ a^m ^ sTir ■\- 7/2" 8. .^^-,2^/+y»--^'"-'^-^"-'^°r-r-^'--^" 170 ELEMENTS OF ALGEBRA. 76. It may be sliown by multiplication (Art. 22) that : {+a)i+b) =ab; (-«)(-&) = ab. (+a)(+b){-\-c) =abc\ {-a){-b){+c) =abc. (+a){+b){+c){+d) = abed; {-a){-b)(-c){-d) = abed, etc. Hence, In an indicated product of any number of factors, all the signs of any even number of factors may be changed without changing the value of the product. Thus, (x-y) (:y-z) = (y-x)(z-y); (w -x) (x- y) (y~z) = {X - w) (x - y) (z - y), changing the signs of the first and third factors. Note. In order to multiply a product containing several factors by a given expression the student must be careful to multiply only one factor of that product by the expression. Thus, in order to multiply both terms of the fraction ; — '-— — ~- by a, we must multiply either a-\-b or c + d and m -\- n or X -}- y hy a. 77. It is often convenient to change the order and the signs of the terms of the numerator or denominator, or both. Thus, Change the order and the signs of the terms of the numerator and denominator of the following fractions : b — a m — n 1. -• 2. y -X ' (jc — h) {x — m) Solutions : 1. Multiplying both terms of the fraction by —1, we have b — a __ (b — a) X —I _a — b y—x~{y~x) X —l~x—y' 2. Multiplying the factor x — m and the terms of the numerator by — 1, we have m — n _ (m — n)x— 1 n — m (c ~b){x- m) (c - b) [(x - m) X - 1] (c--b)(m~x)' Multiplying the factor c — b and the numerator of this fraction by — 1, and since adding a negative quotient is the same as subtracting a positive quotient, we have ALGEBRAIC FRACTIONS. 171 n — m _ (n-m) X — 1 + (n - m) (c-b)(m-x)~'^i(c-b)X-l](m-x)~~ (b-c)(m-x)' Change to equivalent fractions having the letters arranged alpha- betically, and the first letter of each factor in the numerator and the denominator, positive : x — m (b — a){c — a) 3. {b-a)(a~c)(y-x) " (d - a)(c - b)(n - m) Solutions : 3. Multiplying the numerator and the factor y — x by — 1, we have X ~ m _ m — X (b -a) (a- c) (y-x)~ (b - a) (ti -c)(x-y)' Multiplying the numerator and the factor b — a of this result by — 1, we have (6 -a) (a- c) (x - y) (a- b) (a ^ c) (x - y) 4. Multiplying the factors c — a and n — m, b — a and c — 6 by 1, respectively, we have (6 -a)(c- a) (a - b) (a - c) {d - a) (c -6) (n - m) (d - a) (b -c)(m- n) Q;„,n„rlv (a-h)(a-c) (a - b ) ( a - c) oimuariy, ^^ _ ^^^ ^^ _ ^y ^^ - n) ' (a - d) (b -c)(m-n) (d — a) (c — o) (n — m) (a ^ d) (b — c) (m — n) Exercise 66. Change each of the following fractions to four equivalent ones with respect to the signs of letters : ^ 7W — 71 _ a — Jj m VI -\- n ^ a a — b^ m + n — x' a — h -\- x' m — n -\- a* 172 ELEMENTS OF ALGEBRA. Change the following fractions to equivalent ones having m and n positive in both terms : m — a a + m — X a + b — n b — n ' b — m — y^ a — b + m X — m X — m {a — m) (b — m) y — n^ (y — m)(z — n)' (c — m) (x — n)(y — m) ' Change the following fractions to equivalent ones having the let- ters of the terms arranged alphabetically and the first letter of each factor in the denominator positive : . 2x-?> — y 3 — c + ft [m. — a) {2 X — b) (b + a)' (y — x) (m — n) (a — c)' ^ (x — m.)ba 5. r^ T7-7T- xy mn{c — b) (b — a) {c — a) {jj-x)yx cb a {b — a) {z — y) (c — a) {y — x) (n — m)' 78. Fractions having a common denominator are similar. Thus, ^, — =-, and — ^ are similar. ab ao ab 2x 3 5 n^ Example 1. Reduce r — 5, «, and -. — i to similar fractions having the lowest common denominator. Solution. Evidently the lowest common denominator is 20 in^n^x^f the L. C. M. of 5 m^, mn% and 4x^. Dividing 20 m^n^x^ by the denominator of each fraction, and multiplying both terms of each fraction by the quotient each by each, we have ALGEBRAIC FRACTIONS. 173 2x 2x X 4n^x^ 'Sn^x* . 6 m« ~ 5 m* X 4 w» x« ~ 20 m^n*xfi' _3 3 X 20ma:*_ _6()mx*_. 5na _ 5 n» X 5 m^ n» _ 25 m ^ n^ 4x^~4x^ X 5 m^n* ~ 20 m^n^icfi* Example 2. Reduce ^aJ^s^^^is * a:«!4x-5 ^ ^'^^ x44^r + 3 to similar fractions with lowest common denominator. Solution. The lowest common denominator is (x — 3) (x — 5) (x + 1) (x + 3), the L. C. M. of the denominators. Dividing the L. C. M. by the denominator of each fraction, and multiplying both terms of each fraction by the quotient each by each, we have x-1 (a;-l)X(a:+l)(x+3) (x + 3) (x^ - 1) x^8x+15~ (x-3)(x-5) X(x+l)(x+3) ~ (x + l) (x-5)(x2-9)' x+3 (x-f-3) X (x-3)(x+3) (x + 3)g(x - 3) xa-4x-5~ (x-5)(x+l)X(x-3)(x+3)~(x+l)(x-5)(x2-9)' x-5 _ (x-5)X(x-5)(x-3) (x - 5)^ (x - 3) x«+4x+3 ~ (x+3) (x+ 1) X (x-5) (x -3) ~ (x+ 1) (x-5) (x^9)* Hence, in general, To Reduce Fractions to Equivalent Fractions having the Lowest Common Denominator (L. CD.). Find the L.C.M. of the denominators. Then multiply both terms of each fraction by the quotient of the L. C. M. divided by the denominator of that fraction. Kotef : 1. When the denominators have no common factors, the multiplier for both terms of each fraction will be the product of the denominators of all the other fractions. 2. In all operations with fractions it is better to separate the denominators into their factors at once; and sometimes it is also convenient to factor the numerators. 3. It will he observed that the terms of each fraction are multiplied by an expression which is obtained by dividing the L. C. D. by its own denominator. It is not necessary to state how the multiplier is obtained in every expression. 174 ELEMENTS OF ALGEBRA. Exercise 67. Eeduce to similar fractions with L. C. D. : a m X ahe 1 2 5 b' n' y^ mn^ ah' ac he' m + n m — n n a a — n n ah be a c b in da m + 2 n 2 m. — 3 n 5 m — n Sm ' 6 n ' 10 m 7^ * 1 x + 2 x-2 6. x'^-V x^-V x-2' x^-x-2 m •— n m + 2n m^ 1 m + n' m — n m^ — 7i^' a + b' a — h' a^ + b'^ 8m + 2 2m-l 3m +2 ft m — 2 ' 3 m — 6 ' 5 ??i — 10 _ xy m — n 8. Tfix — rmy •\- nx — ny' 2 oc^ — 2xy m n a m + x' m^ + x^' m^ — mx + a^ X y m x^ — xy + y^' x^ + X y + y^' ic* + x^y^ + y^ ^^ x — y x^ + ?/2 y x^ + y^ 11. x^ + xy + y^' x^ — y^' x — y^ 5 12. 13. ALGEBRAIC FRACTIONS. 175 b X (a -f xf - 62' (5 _^ 2^)2 _ ^2» a:« - (a + 6)« ate (c-a)(6-c)' (a-c)(c-6)' {c-a)(c-l) a _ g X -1 -g Sestioii. (^_^)(^_c) - f(c-a) X -l](6-c) - (g-c)(6-c)* ** 3a 4a 5 am n y 3"=^' o^' (a -3)2' n^' wrn:' r=^2 12 3 4 14. 15. (2-a;)(3-aj)' (a:-l)(2-a:)' (a:-2)(l-^)' (a:-l)(a>-2) 1 1 Suggestion, (g - x) (3 - a:) = (x - 2) (x - 3) = ^*^- 16. 17. 18. 19. (m — x) (a; — n) * (a: — ?w ) (a — a;) ' (a: — a) (n — a;) 1 + a: 2 -i- a; (l-a;)(2-a:)(a;-5)' (x - 1) (2 - aj) (3 - a:) (5 - a;) a;- 3 a;- 2 g2 + 4 2 4^=^' a^+a;-6' 9-6a;4-ar»' ir2-a:-6' ^ ar^"* + 1 A2m_ 1 a:*"- 1' a^'" + 42:2.-^_ 3' a^« + 2 ar»'" - 3 ' 79. Example 1. Find the sum of t , -\^ and -. o a n Solution. Multiplying the terms of the first fraction by dn, of the second by bn, of the third by 6rf, and adding the results (Arts. 32, 14), we have a c m _adn hen hdm _adn-\-hcn + bdm h'^ d'^ ~ii~ h(U'^ bd^'^ bdTk "^ hdik ^~ " 176 ELEMENTS OF ALGEBRA. m a Example 2. Subtract - from t n Solution. Multiplying the terms of the first fraction by 6, of the second by n, and subtracting (Art. 19), we have a m an hni an — hm h n bn bn b n 2a-3h ^ Sx-2b Example 3. Subtract — ^^ irom — ^ • Solution. Keducing to similar fractions with L. C. D., we have 3 a:- 26 2 a - 3 6 _ 6ax - 4ab 6ax-9bx 3x 2a ~~ 6ax 6ax 6 ax — 4ab — (6 a x — 9 b x) ~" 6ax ■'J _ b(4:a-9x) ~ Hax T^. -11 p 2x — my Zx — ny Example 4. Find the sum ot a -I '- and b m n Solution. Uniting the entire parts, and reducing to similar fractions, we have / 2x — my\ (^ 3 a!; — nwN , (2x—my)n C3x—ny)m [a+ -)+ [h '-) = « + &+ ^^-^ ^^ V m / \ ^ J '^^ ^^ (2x — my)n — (3x — ny)m, = a-i-b-\ '- ■ '- mn , (2 n - 3 m) a; mn Note 1. If the sign of a fraction is — , care must be taken to change the sign of each term in the numerator before combining it with the others. In such case the beginner should enclose the numerator in parentheses, as shown in the above work. 2^; g x+2 a;-|-l Exampi^e 5. Simplify ^2 + 3^ + 2 " x^--2x-3 " x^-x-6 ' Process. ALGEBRAIC FRACTIONS. 177 lar-6 x-j- 2 x -\- I x2 + 3x+2 x^-2x-3 x^-x-6 2 (x - 3) x + 2 x+ 1 = (x + l)(x + 2) ~ (x + 1) (x - 3) ~ (x 4- 2) (x - 3) _ 2(x-3)X(a?-3) (x-t-2)X(x4-2) _ (x4-l)X(a?+l "(x+l)(x+2)X(x-3) (x+l)(x-3)X(x+2) (x + 2)(x-3)X(x+l) _ 2 (x - 3)^ - (x -f 2y -(x+ 1)3 _ 13-18X (x + 1) (x -h 2) (x - 3) ~ ix+ 1) (x + 2) (x - 3) * Notei : 2. In finding the value of an expression like — (x -f 2)*, the be- ginner should first express the product in a parentheses and then remove the parentheses as above. 3. Sometimes it is better not to reduce all the fractions to the L. C. D. at once. Thus, 14 6 4 1 Example 6. -• — 5 -I rTT + x — 2y x — y x x + y x + 2y 1 1 4 4 6 + iTT-zr- - z — - - zr-r-7. + z x — 2y x-h2y x — y x + y x x + 2y x-2y 4 (x + y) 4 (x - y) 6 (x-2y)(x+2y)'^(x-|-2y)(x-2y) (x-y){x-\-y) (x+y)(x-y)"^x 2x 8x 6 ~ x^ - 4 y2 x2 - y3 _ 2 X (x^ - y^) _ 8 X (x« - 4 y«) 6 - (x«- 4 f) (x^-y^) " (xa - y2) (x3 - 4 y^ + X _ 30xy''-6x« 6 -(x»-4y«)(x«-y2)'^x _ (30xt/'-6x»)x 6 (x2 - 4 y2) (x^ _ yS) - (x2 - 4 y3) (x2 -y*)x'^ X (x^ - 4 y^) (x^ - y«) 24 y* TT . = x(x«-4y^(x»-t/) - Hence, m general, To Add or Subtract Fractions. Reduce to similar fractions with L.C. D.; add or subtract the numerators, and divide the result by their L. C. D. 12 178 ELEMENTS OF ALGEBRA. Exercise 68. Simplify : 2a — 5 S a — 11 b + c a + c a — b 12 a "^ l8 ' T^ "^ T6 97" ' X 2x ^x^^ xy xy^ xP"y^ ^ m n fm + n a + b\ /m — n a — b\ ' ab ac b c'' \ n a J \3?i ^a )' 3 + ^^ 4-am a /5 4 3\ /I 2 3\ 4. + + ^; +-)+ )• n an 6n \m n xj \m n xj 5a-b 7a + Sb _ /2a a - b \ 2b "^ 6^ \b "^ 3 & y * 6. (^,. + ^|) + (3m-^)-(4m + ^). ^ a^—bc ac—b^ ab — c^ 2 a^—b^ b^—c^ c^—a^ i c ac ab ' a^ b^ (? ' 8. (m + n ^ ) — ( 2m — 371 + — V \ mxj \ nxj \5 a; Zy 'omj \6 x 10 3/ 7 m/ a + & b — c c — a ab'^ — b(? — c 0? 11. 5 ab c 1 1 :r4-2 x-2 3 oj— 5 ^—4' 2;— 2 a;+2' 2m(m— 1) 4m(?w— 2)" ALGEBRAIC FRACTIONS. 179 12. 2a;/i — 3&71 2a7n + 'Sbn 1 1 '6vin{ia—7i) 3 7/1 7i (?/i + 7i) * ic^*--4ic + 4 "x-'^+x-G* 13. x + y J^ — y ' x—m X'\-m x 14. 1 m+3 -7 2 3 ... 1 •> "T A 2 10> .J.^ " 2m-3 1 A ™2 1 • m n 2 mn 1 (a + 2xy^ m -i- 71 m — 71 ir? — r?' a — 2x a^ — S a^' 2: 1 1 11 16. xy— if' x — y y' m^— (n + xj^ a^—(m + ?i)^ x^+'Sx^f+y* a^xy-\-ii^ 2 3? x + y x^ — ^ X — y ' x*—y* x^+x?y+xif+y^' ,Q x + 4: , x-hS x+2 3?+ ox-{- 6 a? + 6 x+ 8 3^+7 x+ 12 1 mn m — 71 7n + 71 TTi^ -\- n^ 7n^ — m 71 -\- 71^ ' 90 1 1 X X 7n + X m — X " (m + xf (tti — x)^' 21 __i L_ + ^ 4. _ ^ 8-8ic 8 + 82:^4 + 40^2^ 2 + 2a;* 22 24a; 3 + 2a : 3 - 2 a ; 9-12a; + 4ar» 3 - 2 a: "^ 3 + 2a; ' 00 « + & ?^ + r- r + ^ (6 - c) (c ~ a) ■*" (c - a) (a - ft) ^ (a - I) (h - c) 180 ELEMENTS OF ALGEBRA. 24 ^+^y , x-^2y x^y 4.{x+y)(:y+2y) {x+y){x-VZy) 4.{x+2 y) {x+Z y)' ^^ he ac , ah (c — a) (a — b) (a — b) (b — c) (b — c){c — a)' 26 5(2^-3) 7x _ 12(3a^ + l) • 11(6^-2+ ^-1) 6 2;'^+7ic-3 ll(4:X^ + Sx+3)' x _ y x^y+xy^ x^ + 'f ^ — y^ x^+y^ Q^—i/ x^ — y^ ' a^—xy-\-y'^'~x^+xy-\-y'^' 28. ^" + ^* 1-^ a;3 + ^2 _ 49 ^ _ 49 2:2 _ e ^ _ 7 29 ^ + ^' a; 4- & ^ + 0^ 30. {a — b) (a — c) (b — a) (b ~ c) Suggestion. In finding the L. C. D. it is better to arrange the letters alphabetically. Thus, & a b a X —1 + VI — wi — X = / — jaT N + m — \t:^ — TT7I — \ = etc. {a-b){a-c) ^ {b-a){b-c) ~ (a-h){a-c) ^ [(b-a) X -l](b-c) x^+2x+4: oc^—2x-\-4: x-2a 2{o?-Aax) 3^ x-\-2 2—x ' x-\-a a^—x^ x—a 32 1 1 1 . 1 , ^ (m-2)(x'+2)^(2-m)(^ + m)' 2a;+l 2aj-l 4a: 2 a: -3 a?2 + -. T—o\ — ^--o ...... + 1-4^2' ^_l_4 :x:2_4^+i6'^^64 33. 7 iw T + (a — &) (a — c) {b — a)ib — c) (c — a) {c — h) ALGEBRAIC iRACTIONS. 181 Q C 80. Example 1. Find the product of r and ^ . a , c Solution. Let i = a:, and 3 = y. Multipljring both members of the first equation by h and both members of the second by d (Art. 47, Axiom 3), we have a = bxy and c = d y. Multiplying these two equations together, we have ac = bdxy. Dividing both members of this equation by 6 d (Art. 47, Axiom 4), gives ac _ a c ^ = xy. Buta:i/ = ^X^. a c ac Therefore, T ^ ^ = r-, . Hence, in general, « To Multiply a Fraction by a Fraction. Multiply the numera- tors together for the numerator of the product, and the denominators together for the denominator of the product. Notes : 1. Similarly, we may demonstrate the method when more than two fractions are multiplied together; also, for fractions whose terms are negative, integral, or fractional. 2. Since an entire or muted expression may be expressed in fractional form, the method above is applicable to all casesl Thus, ^a m ^a am a / ,n\ a^/>».n\ am , an r. o Ti'j.u J . -4x3-16x4-15 x2-6x-7 Example 2. Fmd the product of ^ q , o — tt, tts — r= ^ ,, „2 , ^ 2x2-1- 3x+ 1' 2x2- 17x-f 21' and 4 x* - 20 X -f- 25 Process. x2-6x-7 4x«-l 4x=»-l 4 x2 - 20 X -f- 25 4x«-16x-|-15 2 x« -h 3 X -H 1 (2x-3)(2x-5) "2xa-17x+ 21 ^4x2-20x-f25 (x-7)(x-hl) (2x+l)(2x-l) (2x-|- l)(x-|- 1) ^ (2x-3)(x-7) ^ (2x-5)(2x-5) (2x - 3) (2x - 5) (x - 7) (X -H) (2x -f 1 ) (2x-l) _ 2x-l (2x-|- l)(x-f l)(2x- 3)(x-7)(2x-5)(2x-5)~ 2x-6' 182 ELEMENTS OF ALGEBRA. Explanation. Factoring the iiiuneiators and denominators of the fractions, multiplying the numerators together for the numerator of the product, and the denominators together for the denominator of the product, we have the third expression. Reducing the third ex- pression to its lowest terms, gives the result. Notes : 3. Observe the importance of factoring the terms of the fractions first. Also, indicate the multiplication of the numerators and denominators, and divide both terms of the fraction by their H. C. F. before performing the multiplication. 4. If the factors are mixed expressions, sometimes it is better to change them to fractional forms before performing the multiplication. Thus, / ah \ / _ ah \ _ a^ iA _ a^lfi V^ a-b)\ a + b)~a-b a + b~ a^-b^' 2 x^ -\- 3 X 4 x^ Qx Example 3. Find the product of — r—^ — and lo^ + is ' Process. 2a;2 + 3a; 4a:2_6a:_a;(2a; + 3) 2x(2x -3) 4x8 >< 12a;+ 18~ 4x^ ^ 6 (2 a; + 3) _ x(2a: + 3) X 2 a: (2 a; -3) _ 2a:-3 ~ 4 a;8 X 6 (2 a: + 3) ~ 12 x * Exercise 69. Simplify : ^ a2 j2 c2 3a3 2h^ 7c^ ^ ^ ^- Fc^'^c^ aV 4.c^ "^ 21 a^"" 5 ah' ^ r' Sah^ 3«c2 Sad^ Sc-^x^ 2() (^ x 2. 4crf 2hd "^ 9hc ' 5a^y-^ 9a-^y-^ x+1 x + 2 x-1 Za^-x 10^ ^' ^^^=~l^ x^-1^ {x + 2f' 5 ^2a;2-4ic a^ + 3 .r 4- 2 x^^-'Jx-\-\2 ^ m^ - n^ m^n ^ _^ 9 a; + 20 ^ :i-2 _|_ 5 ^ + 6 ' ^3 _ ^2^ ^3^ ^£ X' 6_ ALGEBRAIC FRACTIONS. 183 2/6 ^■\-'f x + y X ^- a^ + 2 3^ ij^ + 1/ s^ - xy -{- 1/ "" a^ - f am fm (i\ m^ 4- w* ^i f _!!!: ^ \ * oTw \a~7wy' m^ + n^ \m—n m + nj m^-\-mn nfi — n^ a^—(a-^b)x-{-ab x^—<? "' m^'^z ^mn{m-\-ny a^-{a + c)x+ ac^ x^-l^' m° — m^ + n^ f.. ^ ^ ^' m^ — m n + n^ m^ + mn + n^ \ m — nj 9. g _ ? + 1 V^2 + ^ + 1) • Suggestion. [e-)-i][(s-)-g-(s-)"-0'- ^^ fx a y h\ (x a V h\ 10. ( + ?--Ix( i + -]' \a X b yj \a X b yj \bc ac ab a J \ a + b + cj a^ •\- ab — ac (a + c)^ — V^ ab — b^ — be a:2--2a;"- 63 ^ a^- + 3a:"-40 ^ ?M^4^^+3 ' r a^> y'"* 1>.r (^'^-?/^"')^ 1 8L Example 1. Find the quotient of -. divided by ^' fl c Solution. Let x represent the quotient. Then t -^ -5 = «. Since the quotient multiplied by the divisor gives the dividend, 184 ELEMENTS OF ALGEBRA. we have x X -. = j^. Multiplying both members of the equation d c d a d ad by - , we have a:X-;X- = TX-, oraj^yX-* •^c' d c c^ he Therefore, ^^^=:-^X-=^. Hence, m general, To Divide a Fraction by a Fraction, invert the divisor, and proceed as in multiplication. Notes: 1. Since an entire or mixed expression may be written in fractional form, the above method is api)licable to all cases. Thus, _^a _ c _^a _ c ^__^c a _ a c _a 1 a ^ ' b"! ' l~l^a~ ~^'' b'^^~l^\~b c~bc' 2. It is usually better to change mixed expressions to fractional form before performing the division. Thus, (_a6\^/ ab \ _ <fi ^ b^ _ a'^ a + b __a^ " T+b) ■ V aT6/ ~ a + 6 ' aTl> ~ ^TTb ^ ~W~ ~ P ' ^. ., ar2-14a:-15, x^-l2x-45 Examples. Dmde ^,_^^^^^ by ^^--^-_^. Process. a:g-14ar-15 . g;2-12a;-45 (x - 15) (x + 1) , (x - 15) (a: + 3) a;2-4a;-45 ' a;2-6a:-27 ~ (x-9)(x + 5) ~ (ar-9)(a; + 3) - (a^-15)(a:+l) (re -9) ~ (a: - 9) (a: + 5) ^ (x - 15) _ (a^-15)Ca;+l)(a:-9) x+ 1 " (a: - 9) (a: + 5) (a; - 15) ~ a; + 5 ' ^. ., a?^ 1 , a: 1 1 Example 3. Divide ^+ibyp--+-- Process. \y^ x) ~ \y^~ y x) ~ xy^ ' x y^ _ {x -\- y) (x^ - X y + y^) xy^ xy^ x^— xy + y^ _x + y __x ALGEBRAIC FRACTIONS. 186 Exercise 70. Divide : 2 a^ 2:1 y , a x^ y"^ Z m , 2 m 6 (g 6 - ^/2) 2 62 2;3_y3 (a; - y)8 •^- a (a +6)2 '^^aCa^-feZ)' a^ -^ f ^ {x + y^' x^-f x-y ^ a^^xy+ip' ^ - 1?' m8+8^m + 2' 2? ^- f ^ 3?-xy-\-y^' ^ 2^:2+13^^15 22:2_^ 11^ + 5 5- ^^_Q by 4a:2_9 -^ 43^2 a:2 4- a: y + ?y2 x + y m m 6. ;;o ^^-H^ by — ^^; -^ ^ by - + -. x^ — xy-)r]rx — y n^ m^ 71 m yj X-^y x-y x+if x-y X y ' X ^ y X + y ^ X — y' x -\- y ^ y x' x^-^ (a + c)x + ac x^ - o? ^- a?» + (6 + ^) a: + 6 c ^ ar» - 62 • a2 4.^>2_^2ft6~c2 « + 6 + c ^- c2-a2_i,2+2a6 ^ 6 + c-a' 10. a;8-^ by a:--; a2-62-c2+2&c by ^^44^^- a:^ -^ a: -^ a + 6+c 11- ^6 . ^6 by -2-7-0; -e— r by 7i6 + a:« ' 7i2 + a:2» fl^6__i ^ a^^a^^a-l ^^ x~^ — x~^. x~i -{- x~^ 12. 0^-8 by 2a:-8 4a^2(x-f-a:-*) 186 ELEMENTS OF ALGEBRA. Exercise 71. Perform the operations indicated in the following and reduce the results to their simplest forms : 7 ic + 6 . a:2 + 6a; \ . . a;2 + lOa; + 24 48* /x^-7x + 6^ x'^ + 6x\ x^ + 10a; + ^* \x^-h 3a; -4 ~ x^ - 8 x^J ^ a;^- 14a; + / x2 - 7 a; + 6 a;2 + 6a;\ a;'^ + 10 a; + 24 Process. (^^2 + 3 a; _ 4 "^ a;3-8a;2j >^ a;^- 14a; + 48 -i [ (^ - 6) (a; - 1) ^ a;(a; + 6) 1 (a; + 4) (a; + 6) ~ [(a; + 4) (a; - 1) "^ x'^ {x - 8) J ^ (a; - 6) (x - 8) _ f a; -6 a;(x-8) 1 (a; + 4) (a: + 6) ~ [a; + 4 ^ a; + 6 J ^ (a; - 6) (x - 8) _ a; (x - 6) (a; - 8) (x + 4) (a; -f 6) _ - (x + 4) (X + 6) (X - 6) (x - 8) ~ '^' a-l a + 1 «2_i X : a + i a — l'a + 1' Va-&~ a + &J * a2-62- \x + y X — y x^ — y^J \x -\- y or — y^J ^2 _ ^ _ 20 x^-x-2 x^-S6 ^ + 1 4. — 5 :t?— X -0-7-r. o X x^-2d x^ + 2x-S^x^-6x ' x^ '\- 5x ?/4 a + h x^—3xy + 2y^ ^ {^ — vf X 7 ; ZTt X ' a^h + ah {x -\- y)^ x^ + y'^ ' ah /a-f Z) « — &\ /« + & a -- &\ max a^ — x^ h c -}- h x c — x ^ mx noy c^ — x^ a^ + a X a — x ny ALGEBRAIC FRACTIONS. 187 8 1 • PM I Mx ^"^ 1. ' x-^y \_2\x + y x — y) x^y + xy^J x^-\-x-2 aP-^5x+4: . f x^+Sx-{-2 x+3 \ ^^' \6x-62 * x^^l^J'^ax+a^' 8a^y^^ 21b"'^^y'^-^ X 6^mH^-a^ (a: - 2)^ , ar^ - 4 ^a' ,^2-4 ^ 8 ?w?i + a * (ir + 2)a* 12. ., o a X — =— i- X '^ 14. (a:* — -jjH-fa; J, by inspection. 15- (p - 2 + ^2) - (? - I)' ^y inspection. 16. ic^— -g — sfa; j \-^ix ), by inspection. 6flg^>g , r 3a(m-7i) . 5 4(c-a;) ^ c^ - a^ ^ m+n * I7{c + x) ' I 21afe2 * ^(m^-n^ij 188 ELEMENTS OF ALGEBRA. 82. A Complex Fraction is one having a fraction in its numerator or denominator, or both ; as, n' , m * n + — n Example 1. Eeduce - to its simplest form. c d Solution. A complex fraction may be regarded as representing the quotient of the numerator divided by the denominator. Hence, a h a ^ c a d _ a d cj^l^d'^'b^'i^'bc' d h a — - Example 2. Reduce f to its simplest form. m Solution. Since the divisor is m, we have 6 \c — b m ac — h 1 ac — h = [a — ~\ -^ m = - x- = I c m cm I I m , m Example 3. Reduce ~ » T » and — to their simplest forms. ^ 1 m n n Process. — =zi-f- = i x — = — m n mm m 1 n Y = wi-r- = m X ^ = mn. n 1 m 1 1 1 n n -r-=~"^~ = ~ X7 = -. Hence, m general. I m n m I m ' & ALGEBRAIC FRACTIONS. 189 To Simplify a Complex Fraction. Divide the numerator by the denominator. Example 4. SimpUfy "*' " ^^' ~ '"' + '^^ m + n m — n m — n m + n Process. m -j- n m — n ~ (m -h ?i)2 — (771 — n)^ ~ 4 wm m — n m 4- n (m — n) (m + n) (m — n) (m + n) 4 m^n* (m — n) (m + n) in n X (m« - n2) (m^ + n^) '^ 4 mn m* + n^ Example 5. Simplify ^ ^ iiJ2 x-y x+y Solution. Multiplying both terms by (x — y) (x + y), the L.C.D. of their denominators, we have 2x1/ Notes : 1 . In many examples it is advisable to multiply both terms of the fraction by the L. C. D. of its denominators at once. 2. If the terms of the complex fraction are complicated, the beginner is advised to simplify each separately. mp Example 6. Simplify ^'+(^+^03:+ mn x^ + {m+p)x + mp X* + (n + /?) X + n p 190 ELEMENTS OF ALGEBRA. Process. mp mn mp v'^+(m-^n)x + m7i x^-h{m+p)x + mp _ {x-\rm){x + n) (x+m)(x+p) n — p ~~ n ~- p x'^+ {n+p)x + np {x + n){x+p) m n {x-\-p) — mp (x+n) _ {x-\-7ri) {x+n) {x+p) _ inx (n ~- p) {x + n) (x+p) n — p ~ {x + m) {x+n) {x+p) n—p (x + n) {x+p) rax X + m Example 7. Simplify T+~x' 1 '- l-x + x'^ + a:2_ 1 1 + X- X Solution. Begin with the complex fraction x^—\ ' '^^"®» a:2 _ 1 x+l , X x'^ i + x-—— = -^, and ^5^n[ = rri- ^'"^"^^^^y i + ^-^r- l +x^ (l + x^){l + x) ,^ l+x^ • ^= ,:8 + ^2+i ,and 1-- x^ l-X + X^+ ——r l-X + X^ + x^ + x^+l Therefore ' 1 +x^ X 1 X x^ + x^+l 1 - X + X^ + -o 7 1 +x- ——- X = - (X8 + X2 + 1). Notes : 3. A fraction of the form in Example 7 is called a Continued Fraction. 4. To simplify a continued fraction, the student should always begin with the last complex fraction in the denominator. ALGEBRAIC FRACTIONS. 191 Exercise 72. Keduce to their simplest forms : a; + 6 + ^ 1 + - a: + - xy ^ j: — 6 wt c m n 1 ' 6 ^ ' mm n ^ X-6 + 1 a;+- + 271 X + o m n X , . 6^ 1 . 1 ni b m + n a-\-b + ^ - + — . ^ a n m n m 4:mn n mm n S m^n a;+la;-l ar8-17a;+72 2 x-1 ./J + 1 , 2:2 ^ 22 a; 4- 120 ^^ + 1 x-V x" - 21 a: +708 x^^ iiTTl a?» + 18 a; + 80 1 + « 4a 6 \a^; aj\x a) l + a6 2a6 a; + a J, + J_.JL - mn mp np f x , 1— A_/ x \—x\ ' m^-(/i + pF ^ Vh^ ~^y \i+^"""^/ 6. m 71 1 1 1 ^ 1 ' 1 ' ^-1 • «.' + J 1 X — x + - 1 + i ^ + — ^- a; a; a; — 1 192 7. ELEMENTS OF ALGEBRA. 1 X-2 1 + l + a; + '1^ 1-x x-2 x-1 x + y X X 1 , 1' _L y^ y X x — 2 2 xy ~ (^- + yf 1+ x+ 1 d + m 1 + ^,y (^' - yf X ax a? «2 X' X X X x^ ir + -0 -0+-+1 a a? 2/^ ^ X — y x" y^ y ^ 10. 11. w? + n^ 2 m m^ — ii^ ' 7?< + n ~m n — m^ ^ m -{• n (m — iif' ' 'IV — n_ m — n 71 + m — n 1 + m n {in — n) n 1 -\- mn m m — n 1 - • m n 1 - (m — n)m 1 — mn ^ fm n\ \n m) 83. Example. Find the third power of t; Solution. Since an exponent shows how many times an expres- sion is taken as a factor, we have 'aJ^\ {aJ^y J^nj - 6" >^ &« >< ftn - (hny ~ h^n ' Hence, To Find any Power of a Fraction. Raise both terms of the fraction to the recjuirecl power. ALGEBRAIC FRACTIONS. 193 Exercise 73. Expand, by inspection, the following: ^(-•-^)' m' [-G)'^a)T / 2aHixi y' f ix + ;/f y r m (x - y) !^ r (r+,,)(x-,,) y r («-i)-»(«-5)n « /a^a-5)i\w *• L "« + » J' L (2«+3t)i J' ^ ^ J «-C4?)U©-©T= + 1 arff X wi ai Jx 7«^ 84. Example 1. Find the rth root of 6" Solution. Since the rth power is found by takinp^ the numerator and denominator r times as a factor, the rth root is found by taking the rth root of each of its terms. The operation is indicated by dividing the exponent of each term by r. Thus, 13 194 ELEMENTS OF ALGEBRA. 5«4- Illustration, y 243^ =^ ssTs^^isTs - 3-p • ^®^c®» To Find any Root of a Fraction. Take the required root of each of its terms. Example 2. Find the square root of^2-"2 — aa;+j + — + ^- Arranging according to powers of a, we have X a Process. First term of the root squared, First remainder, First trial divisor, a^ a First complete divisor, « + - - times first complete divisor, Second remainder, Second trial divisor, 4 + ^ + ^^-"^-^ + ^^ x^ a^ a2 + 2a X 2 a X Second complete divisor, a'^ + -— — - — - times second complete divisor, a8 + a2 X^ X rc2 -ax- -2 + a2 -aa;-2 + a2 Note. If we take — |- for the square root of ^ , we shall arrive at the result — -p: f- -- . 2 a; a ALGEBRAIC FRACTIONS. 195 Exercise 74. Find the values of the following expressions : 1- Vy^286' V a^ ^ V 343 ar^* ^ ^243 ^.25" j • f ni^n^ \^ ( Z2a^\\ ( (SA.m^n^x^ \\ \ a^^ ) ' \ b'^ ) ' \ 12oaH^7/y ' Find the square roots of : ^, .. ^^ . 4 . ^^^ . ^ Miscellaneous Exercise 75. Reduce to lowest terms : h(b-ax) + a(a + bx) 2^-9 a^ + 7ix^ + 9 x-S (b-axf-h{a + bxf' a^ + Ta:^- 9 ^ - 7a;+ 8 * 21 x^ y^ - S5 i/^z- 12 x^z + 20 xyz^ l8x^z^-21x^y-S0y2^+S5xfz' 40r ^y*-?>2 2^ yz^ -5i/^z^ + 4xs^ 40:^28- 36a:*22'-5/2 + 4oa^2/8* 196 ELEMENTS OF ALGEBRA. ^3~6a;2-37;r + 210 ^ a;4»+10 ^""^ 35 rg2"+ 50 a;^+ 24 ■ 0^+4x^-^7 X-2W ic3»+9^2n_|.26a:" + 24 Find the values of : -_^2a;2/^2;^, . - ^a — ic 0. o x^ -] o when x = 4, v = -k, z = 1: r: when 2; = a + b „ x^ + y''^ — z^ + 2xy . ^ . ^ 2; — a ,x' — 6 , ft2 X X 7. — ^^ when x = 7; - + b a a—b a b—a a+b a^ (b — a) when X = , ' , — ^ - b (b + a) , ^ ... ^ — 2 a + b a + b 8- < ^ - + a-2b ^^^'^ ^ ^ ~2~ • (X — «V x X — b ) X 9. i7^[_r + V ^ 2 J when x=^, ?/ = 1. 2ab-\-2bc^-2ccl-^2ad ' y(a_c)(J+c)+6(-c-&)(a-2&) when « = 3, 5 = 1, c = - 2, t? = 6. g^ + g c 4- &2 ^ {4^ai^yij^ ^^\ c ^2 — « c + 52 V'4a& — 52-2^ ~ a + 6 + cH-fZ when a = 4, & = 3, c = 1, c? = 7. Divide : 12. !»J-^by™-x;^+gby?+2^. ALGEBRAIC FRACTIONS. 197 1 , \ t:^ m^ . X m 13. m* i by i?i ; -4 r by - H 14 a^ + ^byx + -; a;3 ^ ^ + ^ _ _ by ^ - - . 15. a2_j2_,2_26cby ^44^'- 16 • ^+i-K^^-^0-''(^'^-^)'^^-'^ Factor : Simplify : 20. 3x-{y + [2..-(y-.)]} + i + |^^. a;— 1 x—\ a; + 3 a: + 3 "3"*^^^ . 7 ~^T4 ^■^- a; + 2 a; 4- 2 * a;— 2 x—2' '~T'^x^ 3 '^0^1 a-\ l-\ c-\ 3a5c a h c ' be -h ac — ab la.1 1 a 6 c 198 ELEMENTS OF ALGEBRA. 23 ^ , ^ I ^^ + ^ , 9^>2-(4c-2 a)2 Uc^-{2a-uy^ 4.a^-{U-4.cf ' (2a+3&)2-16c2 + (36 + 4c)2-4ti2 + (2a+4c)'^-9&2 \in/' j\m—n J \n^ J \m^-\-m7i+n^ J h a + 1 X -{- a 1 X — a i_L^ 27. ^IZ+f! + 5^Z±Z; ^ X («^-5«). 1 a + a; 1 a — x' h ^ ' a 0^ + Q(^ ^ <x2 _}_ ^2 ^ h a — h — c h — c — a c — a — h a^—ac — ab + bc b'^—ab — cb+ac c^—bc—ac+ab X X 11 1 30. 2/ + r ALGEBRAIC FRACTIONS. 199 31. / 3 2: + 3:3 X2 yi-f 32:V 9^ _ 33 - ar^ 3 ^ 32:2+1 ahc a:3_3a:'^ 2:^ (a^-a:)^ hc^ ac ab S-a-b + c 62 a-16-2 a^b-^ (a-^b-^y a ^b — c * a^ ab~^ aH~^ / a-2 6-^ Y '^2:'"+?/" '^2^^'"+?/2'* a-U^ &Ki ai(^ 2:-+y" x^+y^ jic-i 10 2; 7/ -3^2+ 10a:-3y _^ lQ2:-3y _3_ 15i/^+102;//2 + :30y+202:y ' 452/ + 302:y ■*" y+2 ' - (f-)(i^r')H"-')(Sa?-) 35 ^ + ^'' ^ + ?^^\.f + y') ■ \ff x) (x + y)^ — xy (x — yf-\-xy r g<-?/ ^ rt2+a?/ 1 To^-oV g^-2qg.v+fl27/n 37. * 1 Sax — 5b y ax '^by'^ 1 ^""3a2;-2fey 38. -i-^+-l-+ 1 1 + i— 1 +-T— 1 + -t z X -\- z X -{■ y 200 ELEMENTS OF ALGEBRA. 39. (a + 2> + .)^- + ^ + -j- ^^^ (c - a) {a-h)'^ {a- h) {h - c)'^ {h - c) (c - a)' fa^—ij^ lOx^—lSxy—Sy^ 2a:^ + xy^+xy+y^\ \x^—jf l<)oiP'—Zxy—y^ 2x^ + xy—%f' ) xy — y'^— 2x+2y ~ 2a;-?/ (a + hf - 6-3 {h + cf - g g (ci + c)3 - 63 44. 45. (a + i) — c h -\- c — a a ■\- c — h 11 1 a(a — &)(a — <?) b(b — a)(b — c) abc 2a + n a + b + 71 m + n—a am+ab—bm—o? ab+bm—am—b^ m^—bm—am+ah 46. -7 rc^ X + TT, TT^ -X + a(a—b)(a — c) b(b — a){b — c) c{c—a)(c—b) Queries. Why does changing the sign of one factor of either term of a fraction change the sign of that term ? Will it change the sign of the fraction ? Why 1 When the denominators have no common factors why multiply both terms by the product of the denominators of all the other fractions 1 Why does the process of reducing to forms having a common denominator not change the value of a frac- tion ? How prove the methods for addition, subtraction, multiplica- tion, and division of fractions 1 FRACTIONAL EQUATIONS. 201 CHAPTER XV. FRACTIONAL EQUATIONS. 85. Example 1. Solve -j^ - 147^315 = "gT 30~ J_ ■*"105* Solatioii. Multiplying each member by 210 (the L. C. M. of 15, 21, 30, and 105), transposing and uniting like terms, we have — j = 5 + 30 X. Multiplying each member of this equa- tion by X— 1, transposing and uniting like terms, we have 25a; = 100. .-. x = 4. Proof. Substituting 4 for x in the given equation, we have 6-5X4 7-2X4" _ 1+3X4 _ 10 X 4 - 11 1 15 "14(4-1) ~ 21 30 "^105* or, — ^^ = — ^yV' which is an identity. Hence, To Clear an Equation of Fractions. Multiply each member by the L. C. M. of the denominators. o c, 1 2x+U 2fx-l x-l Example 2. Solve =— ^ - ^ = -^ • 5 50 z — 10 2^ Proceas. Multiply by 5, 2 x + 1^ - loT^ = 2 x - 1. Transpose and unite, - r^ — — ^ = — 2^. Clear of fractions, - (2^ a: - 1) = - 25 x + 5. Transpose and unite, 22.6 x = 4. .*. x = ^W* 202 ELEMENTS OF ALGEBRA. Note. In solving a fractional equation, where some of the denominators are simple and some are compound expressions, it is better to multiply each member of the equation by an expression which will remove the simple denom- inators tirst, then transpose (if necessary) and unite like terms. Similarly remove the compound denominators of the resulting equation. Exercise 76. Solve the following equations : ■ X ^ Vlx~ 24' 2 '6x ~ ^ ~ 'Ix ' 6a;+13 _ 32:4-5 2x 2^-5 x-?> _4a:-3 ^ 15 5x-25 ~ 5 ' 5 "^ 2i^=^ ~ 10 i^- 9 2^+5 8^-7_36^+15 lOJ 14 ^6^+2" 56 "^l4" 4a:+3 73;-29 _ Rrr+19 3,x+2 _ 2a:-l ^ a; 9 "^5:^-12" 18 . ' 6 3a:-7 2* ^ \^x-Tl ^ I+I62; ,. 101-642; ^- ^9^^6^+2^ + -^4-^^^^-— 24 6. 18 2^+10 72 a; + 30 _ 20.5 16 2; - 14 42 168 ~ 42 18aj+ 6 1 2 _2;+2 4(2;4-3) _ 82;+37 72;-29 ^- 2"^'2:+2~ 22; ' 9 "" 18 52;- 12 ^ 2 2: + 8r} _ 13 2; - 2 , ^ _ 7^ _ 3^+16 ^ 9 17 2^-32 ' 3~ 12 36 g+ 1 22:-4 22;- 1 x-2 x-4: T5~ 72;-16~ 5 ' .05 -0625 FRACTIONAL EQUATIONS. 203 86. Frequently it is better to unite some of the terms before clearing the equation of fractions. Thus, X ^^ ~ 3 16x4- 4.2 23 Example 1. Solve -^^y + g^.^^ = ^ + ^Tl * X ^^~3 23 16X+4.2 ^ Process. Transpose, — ^ - ^q-[ + -3^+2" = ^- ,, . ,, ^~3 I6X+4.2 ^ Unite like terms, jTTj" H — g , « = 5. Free from fractions, 4+V-x-a:^+16x2+20.2x+4.2 = 15a:^25a:+10. 1.6x Transpose and unite, — g— = 1.8. .-. a; = 3f. Example 2. Solve -^—^ - jqjg " ^a^Ti = 0- Process. Multiply by x^ - 4, (x + 2) - (x - 2) - (x + 1) = 0. Simplify, -x+3 = 0. .-. x = 3. Notes : 1. If a fraction is preceded by the — sign, in clearing the eqiiation of fractions, care must be taken to change the sign of each term of the numerator. In such case it is convenient to enclose the numerator in parentheses before clearing the equation of fractions. 2. The student should be careful to observe that he can make but two classes of changes upon an equation without destroying the equality : I. Such as do not affect the value of the members. II. Such as affect both members equally. Thus, in the above process, the first operation affects both members equally; and the second, that of uniting like terms, does not affect the value of the members. 4 2 5 24 Example 3. Solve ^-j^ - ^^^^ = ^-^ - ^-^ . Solution. Transposing, ^ - g^ = ^ - 2F+2 ' 204 ELEMENTS OF ALGEBRA. Simplifying each member separately, we have 3 _ 11 1 2 (x + 3) 2 (a: + 1) ' "^ a; + 3 - 2 (a: + 1) Clearing of fractions, we have 2 (a; + 1) = x + 3. r. x=l, -c^ A oi ^ — 4 x — 5 x — 7 x — 8 Example 4. Solve r x-5 x-Q x-S x-9 Solution. Reduce the fractions to mixed expressions, 1 1 1 1 ^ , . , or ^-3-^ - ^^^^ = ^-jg - ^^^ • Reducmg the terms in each member separately to common denominators and adding, we get - (a:-5)(a;-6) =" ~ (a: - 8) (a; - 9) ' ^^^^""^ *^^« ^^"^^^^^ «f fractions, we have —{x — 8) (^ — 9) = — (x — 6) (x — 6). Simplify- ing, transposing, and uniting like terms, — 6 a: = — 42. .'. x = 7. (2 a; + 3) a; J^ 2a;+ 1 "^ 3a; dx A- 3^ X Process. Reduce ^^ , i to a mixed expression, Transpose and unite, - ^j:^ == - 3^ ' Clear of fractions, — 3a; = — 2a:- Therefore, a; =.- 1. -n. r. , 5a;-64 2a:-ll 4a;-55 Example 6. Solve a:- 13 x-Q x-14: x-1 Process. Reduce the fractions to mixed expressions, FRACTIONAL EQUATIONS. 205 Simplify each member separately, 7 (x - 13) (x - 6) ~ (2: - 14) (x - 7) Divide by 7 and clear of fractions, x2 - 21 a; + 98 = a?2 - 19x + 78. Therefore, x = 10. Exercise 77, Solve the following equations : 12 1 29 x + 4 x+6 a; ' 12a; 2.4* 32;-8 3a:-7 3a:+l a;-2 6a:+l 2 a; - 4 2a;-l 3(a:-2) a;-l' 15 7a:-16 5 x-\-25 ^ 2x + 75 5 4 _ 3 a;-5 "" 2a:-15' 1 - 5 a: "*" 2 a; - 1 "" 3a: - 1 6a;+8 2a;+38 _ x^^x+1 a^+x-{-l _ ^ 2a:+l a;+12 ~ * a;-l "^ a: + 1 " ^^' ^7_2^--15 1 J 2 1__ a:+7 2x'-6"*"2a;+14"~ '1-a: 1+a: l-x^~^' 3 30 3.5 4 - 2 X- 8 (1 - a;) 2 - ./; 2 - 2 a; 6^-7i l + 16a: , 121-80. ^' 13-12a:+'^^+ 24 ^ ^^^ 3 a;— 1 a; — 5 a; — 4a;--2 aj — 2 a; — 6 a;— 5 a* — 3 5a;-8 6a;-r44 10a;-8a;~8 ' a;-2 ar — 7 a; — 1 "" a; — 6 ' 206 ELEMENTS OF ALGEBRA. x-l x+1 _ 2(x^ + 4x-tl) ^^ • x-2'^ x + 2~ {x+2f , , 30 + 6 rr 60 + 8 a^ ^ . 48 11. ; — H ; — rt — = 14 + X + I X + O X + 1 .6a:+.044 .5^--.178 _ .3a^-l _ .5 + 1.2 a; ■^^' A .6 ^-^^^ .5x-A~ 2x^1 2x-Z Ax -.6 1-lAx _ .7{x-l) 13. .3ic-.4 .06;:c-.07' x + .2 .1 - .b x 87. A Literal Equation is one in which some known number is represented by a letter; as. X X Example 1. Solve — f- m n — m m -jr n Process. Clear of fractions, x {n"^ —m^)+x (m^ +mn) = m^{n- m) Unite like terms, {n^ +mn)x = m^{n — m). m^(n—m) Divide hy n(n + m), x = ^^^^^_^^^y Example 2. Solve (x-m) (x-n) — (x-n) {x-a} = 2(x-m) (m-a). Process. Simplify, transpose, and unite, Sax — 3mx=: — 2m^+ 2am ~ mn -\- an. Factor, 3 (a --m)x = (a — ?«) (2m + n). 2m + n Divide by 3 (a — m), x = ^ — • a2-3&x ,„ , 6 6a:-5rt2 Examples. Solve ax ab^ = ox-\ -^ a 2a bx + 4a 4 Process. Clear of fractions, simplify, transpose, and unite, 4a^x-3abx = 4a^b^- 10 a^. Factor, a (4 a - 3b) x = 2 a^ (2b^ - 5). Divide by a (4 a - 3 6), • x= ^\_.^f^ ' FRACTIONAL EQUATIONS. 207 _, ax ~b bx — a a — b Example 4. Solve , . — l^ , ,. = /„ ^ , h\ /k ^ ^ „\ ' ax -{■ ox + a {ax + o) {ox + a) Solution. Reducing the terms of the first member to mixed ex- / 2b \ f 2a \ a-b pressions, we have [l - -^-^ j - [l- ^^^j = (^^^^^^^^^^^ • Uniting like terms and reducing the fractions to a common denominator, adding and factoring their- numerators, we have 2(a-{-b) {a-b)x a-b ^,, . ^ , . 7 . iv /I. — ; — 7 = 7 , .V .. — ; — ; . Clearing of fractions, {ax + b) {bx + a) {ax + b) {bx + a) " * 2{a + b) {a — b) X = a — b. Therefore, x — ^ . . v • Notes: 1. Example 4 may be solved by clearing tlie equation of fractions. The solution is presented as an expeditious method. 2. If the student cannot readily discover a special artifice, be should clear the equation of fractions at once. 3. Known terms are called absolute terms. Thus, in the equation mx^ •\- nz -\- a = 0, a is called the absolute term. a -i- b a b Example 5. Solve ; = 0. x — c X — a X — b Process. Clear of fractions, {a-\-b){x-a){x-b)-a{x-b){x-c)-b{x-a)(x-c) = 0. Simplify, transpose, and factor, x{ac + bc - a^-b^) = ab{2c -a-b). Tx. ., , , « .„ ab{2c-a-b) Divide by a c + 6 c - a* - 62, x = 7^; 5 — A* ^ ' ac + bc — a^—b^ _fl &(a + 6-2c) °' ^~a2 + 63-c(a + 6)' Exercise 78. Solve the following equations : 10. e a 6 1 X a ^x 2. \0hmx — ^an = 2am — hhnx\ = r* a X X 208 ELEMENTS OF ALGEBRA. ^ 7n? n 4:71^ m a h „ ,« X A X 4: ox ax 4.-|.(.-«)-(^-±^J=^(.-|). 5. ^^ - ^^ +2 = 0; (x-a)(x-h) = {x'-a-hf. ^ a{b^x + a^) ax^ 2fx A Zfx \ 6. -^— V ~ aca; + ,-; -- + 1)=-- — 1). hx b S\a ) 4\a / 3 ah — x^ 4:X — a c x^ — a a — x 2 x a ^ ^ ^ ^ ^^ w^.^ l^ ^ ^ ' c hx ex ' hx h h ^ X — m 0? — mx — V? ^ n? L — 1 • VI mx — n^ mx — n^ Miscellaneous Exercise 79. Solve the equations : X 07+1 ^ — 2x'^ac hc_ , 9 ^ ~1> 1 - 9aj' h~x~'^~^ '^ ' ax+h Sh ^ a^x^ + h^ ax — h a X + h~ a^x^ — b^' X X ■\- \ _x — ^ X — ^ ' x — 2 X — 1 X — ^ X — 1 ' 2(2a;+3) 6 bx+\ 4 1 63-9^ 1-x 2% -Ax FRACTIONAL EQUATIONS. 209 5 1 I 1 1 0. a (6 — a;) b{c — x) a(c ^ x) 6. (2a:--^)rar+^^ =4a:^^-a:Vj(a--4a;)(2a + 3a:). 17 __ . _ 105 +10a; _ .^ ^* a; H- 3 ^ ~ 3 a: + 9 8. (.^3)^_^^^) = 7.-(3.-?i^)). a?— a a + a; 2aa;_ 1 ^ 1 __ a — h ' a—b a -{• b a^—l^~ * x — a a;— 6 x^—ab 10. 3— + — j — = 2 a;; -=c(a — ^)) + -. a; — la; 41 x ^ ^ x _ a:+ 2 , a:- 7 a: + 3 x - ^ 11. h ■= — — — z- = 7 • a; X — o X + 1 X — 4: 135 a; - .225 .36 .09 x - .18 12. .15 a; + 13. .6 ~ .2 .9 x—a x—a—1 x^b x—b—1 a;-a — 1 a? — a — 2 x—b — l x—b — 2 ^ . SO a — bx 9 n — ax 6 m — nx 14. = 5 ^ = 0. ,^ 4m(a2-5./2) ^ 5 m (J^ - 2 a;) 8a: 4 X — np X — mp X — mn 16. p = p. ^ m n V 14 210 ELEMENTS OF ALGEBRA. 3 & (a; — a) a; — &2 ^ & (4 a + c a?) 5 a 15 6 ~ 6a mx — n mx -\- n c^ — Sdx(P+2cx X X ' c^+odx d^—2cx~~* m ~~ n n ',n m x 7n(x—m) x(x + m) mx x m x{x + m) m{x—m) m^—x^ Queries. Upon what principle is an equation cleared of fractions ? How is it done '? Why change the signs of the terms of the numera- tor of a fraction, preceded by a minus sign, when clearing of fractions '? Upon what principle (give four different explanations) may the signs of all the terms of an equation be changed ? Exercise 80. 1. The second digit of a number exceeds the first by 3 ; and if the number, increased by 36, be divided by the sum of its digits, the quotient is 10. Find the number. Solution. Let x — the digit in tens' place. Then a; + 3 = the digit in units' place, and 2 a; + 3 = the sum of the digits. Therefore, 10 a: + a: + 3, or 11 a: + 3 = the number. lla:+3 + 36 Hence, — ^ — r-5 — = 10. .-. a:= 1. lla:+ 3 =14, the number. PROBLEMS. 211 2. The first digit of a number is three times the second ; and if the number, increased by 3, be divided by the differ- ence of the digits, the quotient is 17. Find the number. 3. The first digit of a number exceeds the second by 4 ; and if the number be divided oy the sum of its digits, the quotient is 7. Find the number. 4 The second digit of a number exceeds the first by 3 ; and if the number, diminished by 9, be divided by the sum of its digits, the quotient is 3. Find the number. 5. A can do a piece of work in 7 days, and B can do it in 5 days. How long will it take A and B together to do the work ? Solution. Let x = the numler of days it will take A and B to- gether. Then - = the part they do in one day ; but = = the part A can do in one day, and e = the part B can do in one day. Therefore, = + ^ = the part A and B can do in one day. 7 o Hence, - = ^ + ^. Therefore, x = 2\^. 6. A can do a piece of work in 2 J days, B in 3 days, and C in 5 days. In what time will they do it. all work- ing together ? 7. A can do a piece of work in a days, B in 6 days, C in c days. In what time will they do it, all working together ? 212 ELEMENTS OF ALGEBRA. 8. A and B together can do a piece of work in 12 days, A and C in 15 days, B and C in 20 days. In what time can they do it, all working together ? 9. A and B together can do a piece of work in a days, A and C in 6 days, B and C in c days. In what time can they do it, all working together ? In what time can each do it alone ? 10. A tank can be emptied by three pipes in 80 min- utes, 200 minutes, and 5 hours, respectively. In what time will it be emptied if all three are running together ? 11. A sets out and travels at the rate of 9 miles in 5 hours. Six hours afterwards, B sets out from the same place and travels in the same direction, at the rate of 11 miles in 6 hours. In how many hours will he overtake A ? Solution. Let x — the number of hours B travels. Then x + 6 = the numher of hours A travels; also, I = the numher of miles per hour A travels, and i^- = the number of miles per hour B travels. Then, y^ x = the number of miles B travels, and I (a: + 6) =r the number of miles A travels. Hence, V" ^ = f (^ + 6). Therefore, x = 324. 12. A man walked to the top of a mountain at the rate of 2 miles an hour, and down the same way at the rate of 3^ miles an hour, and is out 13 hours. How far is it to the top of the mountain ? 13. A person has a hours at his disposal. How far may he ride in a coach which travels b miles an hour, so as to return home in time, if he can walk at the rate of c miles an hour ? PROBLEMS. 213 14. In going a certain distance, a train travelling 55 miles an hour takes 3 hours less than one travelling 45 miles an hour. Find the distance. 15. The distance between London and Edinburgh is 360 miles. One traveller starts from London and travels at the rate of 5 miles an hour ; another starts at the same time from Edinburgh, and travels at the rate of 7 miles an hour. How far from London will they meet ? 16. The distance between A and B is 154 miles. One traveller starts from A and travels at the rate of 3 miles in 2 hours ; another starts at the same time from B, and travels at the rate of 5 miles in 4 hours. How long and how far did each travel before they met ? 17. The distance between A and B is a miles. One traveller starts from A and travels at the rate ot 711 miles in n hours ; another starts at the same time from B, and travels at the rate of b miles in c hours. How long and how far did each travel before they met? 1 8. If it takes m pieces of one kind of money to make a dollar, and ?i pieces of another kind to make a dollar, how many pieces of each kind will it take to make one dollar containing c pieces ? 19. The denominator of a certain fraction exceeds the numerator by 6 ; and if 8 be added to the denominator, the value of the fraction is J. Find the fraction. 20. A can do a piece of work in 2 m days, B and A gether in n days, and A and C in m + ^ time will they do it, all working together ? together in n days, and A and C in m + ^ days. In what 214 ELEMENTS OF ALGEBRA. 21. In a composition of a certain number of pounds of gunpowder the nitre was 10 pounds more than ^ of the whole, the sulphur was 4^ pounds less than J of the whole, and the charcoal 2 pounds less than ^ of the nitre. Find the number of pounds in the gunpowder. 22. A broker invests | of a certain sum in 5 % bonds, and the remainder in 6 bonds; his annual income is $180. Find the amount in each kind of bond, and the sum. 23. A broker invests — th of a certain sum in a % bonds, n and the remainder in c % bonds ; his annual income is b dollars. Find the amount in each kind of bond, and the sum invested. 24. At the same time that the west-bound train going at the rate of 33 miles an hour passed A, the east-bound train going at the rate of 21 miles an hour passed B ; they collided 18 miles beyond the midway station from A. How far is A from B ? 25. A person setting out on a journey drove at the rate of a miles an hour to the nearest railway station, distant h miles from his home. On arriving at the station he found that the train had left c hours before. At what rate should he have driven in order to reach the station just in time for the train ? 26. A merchant drew every year, upon the money he had in business, the sum of a dollars for expenses. His profits each year were the nth. part of what remained after this deduction, but at the end 3 years he found his money exhausted. How many dollars had he in the beginning ? SIMULTANEOUS SIMPLE EQUATIONS. 215 CHAPTER XVL SIMULTANEOUS SIMPLE EQUATIONS. 88. Simultaneous Equations are such as are satisfied by the same values of the unknown numbers. Thus, 3 X + y = 9 and 5a: — 2y = 4 are satisfied only hy x = 2 and y = S. Elimination is the process of combining simultaneous equations so as to cause one or more of the unknown numbers to disappear. This process enables us to fonn an equation containing but one unknown number. The equation thus formed can be solved as shown in the preceding chapter. Hote. There are only three methods of elimination most commonly used. Elimination by Addition or SnbtractioiL 89. Example 1. Solve the equations : 5 3a; -5?/ =13 (1) ^ l2x + 7y = S\ (2) Hote 1. The abbreviations (1), (2), (3), etc., read "equation one," "equa- tion two," etc., are used for convenience to distinguish one equation from another. Solution. To eliminate x we must make its coefl&cients equal in both equations. Multiplying the members of (1) by 2, and those of (2) by 3, we have 5 6 X - 10 y = 26 (3) i6a; + 21y = 243 (4) 216 ELEMENTS OF ALGEBRA. Subtracting the members of (3) from the correspojiding members of (4), we have 31 y = 217. .'.y = 1. Substituting this value of y in (1), we obtain 3 a; - 35 = 13. .-. x= 16. VerifiGation. Substituting 16 for x, and 7 for ^ in (1) and (2), we have 548-35 = 13 (1), we nave "[gg, 49^31 (2), , , identities. 32 + 49 = 81 (2), Votes : 2. In this sohition we eliminate x by subtraction. But suppose we wish to eliminate y instead of x. Multiply (1) by 7, and (2) by 5, then add the resulting equations, and we have 31 ic = 496. . •. ic = 16. This value of x substituted in (1) gives y = 1. 3. When one of the unknown numbers has been found, we may use any one of the equations to complete the solution, but it is more convenient to use the one in which the number is less involved. 4. It is usually convenient to eliminate the unknown number which has the smaller coefficients in the two equations. If the coefficients are prime to each other, take each one as the multiplier of the other equation. If they are not prime, find their L. C. M., divide their L. C. M. by the coefficient in each equa- tion, and the quotient will be the smallest multiplier for that equation. Example 2. Solve the equations : 515^ + ^7^ = 92 (1) ^ ( 55 a: - 33 7/ = 22 (2) Solution. Multiplying the members of (1) by 11 (the quotient of 165 divided by 15), and those of (2) by 3, we have 5 165 a; + 847 y = 1012 (3) Xl^bx- 99 2/= 66 (4) Subtract the members of (4) fron the corresponding members of (3), 9461/ = 946. .-. ?/= 1. Substitute this value of y in (1), 15ar+77 = 92. .-. a; = 1. Proof. Substituting 1 for x, and 1 for ?/ in (1) and (2), we have 5 15 + 77 = 92 (1) 1 55 - 33 = 22 (2) Hence, both equations are satisfied for a: = 1 and 2/ = !• Example 3. Solve the equations : 5 ^7 a: - 12 !/ = 289 (1) ^ ( 55 a; + 27 2/ = 491 (2) SIMULTANEOUS SIMPLE EQUATIONS. 217 Process. Multiply (1) by 9, 693 x - 108 y = 2601 (3) Multiply (2) by 4, 220 x+lOSy= 1964 (4) Add (3) aiid (4), 913 x = 4565. .-. x = 5. Substitute this value of x in (2), 275 + 27 y = 491. .♦. y = 8. Prool Substitute 5 for x, and 8 for y in (1) and (2), and we have J 279 = 279 (1), .^^^^^ \ 491 = 491 (2), Let the student supply the method from the solutions. Exercise 81. Solve the following simultaueous simple equations 1. |3a; + 4y=10. Ux"+ y= 9. 8. (Jy + Ja; = 26.» (fy + |.; = 25. 2. Sx- y = 34. a; + 8 y = 53. 9. ( .25 x + 4.5y = 10. 1. 75 y-. 15 a; = .9. 3. 4 10 a: + 9y = 290. 12 2;-lly = 130. 7 y - 3 a; = 139. 2x + 5ij= 91. 10. J 3^ 2 '' l2 + 3 = ^- 5. {6x-5y = -7. \ 10 a; 4- 3 7/ = 11. 11. ( .5 a: + 2y= 1.8. 1 .5 y - .8 a: = .08. 6. 9a;-4y = -4. 15 aj + 8 y = - 3. 12. (7a:+^y = 99. 1 7 y 4- j a: = 51. 7. 9 y 4-2 a; =15. 4y4-7a;= 3. 13. r Jaj4- 3y = 22. 1 l\x-\y=20. ♦ Clear of fractions first. 218 ELEMENTS OF ALGEBRA. Elimination by Substitution. 90. Example. Solve the equations : HaJ + 32/ = 22 (1) ^ l5x-7y= 6 (2) Solution. From (2), x = — — -^ (3). Since the equations o are simultaneous, x means the same thing in both, the substitution of this value of x in (1), will not destroy the equality. Hence, 4/ — F~^) +3^ = 22. Clearing of fractions, transposing, and uniting like terms, 43 2^ = 86. .'. y = 2. Substitute this value of y in (3), x = 4. Let the student supply the method. Exercise 82. Solve by substitution : 1 2. 3. x+Sij^U. ^- l^x + iy = 7. 7 x + 4:y = 29. {S7J + 4cX = SS. Sx+ ?/ = ll. \5x+62j=61. l^V-^x = 21. l3+2 = l- I .08 2/ - .21 a = .33. ( 3 y - 4 a; = 1. I .7z + .12y= 3.54 I 3 a; - 2 ?/ = 1. " ~ *• 10. I 11 -^ ^-- = 0. ^ = 1. SIMULTANEOUS SIMPLE EQUATIONS. 219 Ml2/-7^ = 37. (10a: = 9 + 7y. ^^- |8y + 9a: = 41. U2/ = 15a!-7. 12. < aud verify. Elimination by Comparison. 91. This method depends upon the following axiom : 6. Things equal to the same thing are equal to each other. Example. Solve the equation . )^^-^y=^ 0) ^ (7x-4y = 8^ (2) Solution. From (1), x = i±A^ (3). From (2), x = ?i±l^. Since these equations are simultaneous, x means the same thing in both, — ^ = -^ ^ . Solving for y, we have y = 4. Sub- 7 1 + 20 stituting this value in (3), x = — tj — = 3^. Let the student supply the method. Exercise 83. Solve by comparison : 5a;+6y = — 8. j6x+l5y = — 6. 3x + 4y = -5. (6x+ l5y = -{ I 8 a; - 21 y = 74 \}x-{-iy = S. (-^x + 3y = 51. rSy -.7x =.4. ' \7x+2y = 3. I .02 y + .05 a: = .2$ 12^-7^=17. • U2:+8y = 20. ^ 220 ELEMENTS OF ALGEBRA. (l.lx -l.Sy =0. ^- t.l3:r- .11 7/ = .48. \l + 0^ = 3 2/ -23. ^^1 5 2 ~'^* 12. -{ ^ 1 = 42. r.30^-.772/=:-2.95. h^ 4.y^4o 1 .20 a: +.21^=1.65. ^8"^ 9 92. Each of the equations should be reduced to its simplest form, if necessary, before applying either method of elimination. Notes : 1. An expeditious method, for the solution o^ particular examples, is that of first adding the given equations, or subtracting one from the other. 2. Usually, in solving examples of two unknown numbers, it is expedient to find the value of the second by substitution; but this is by no means always so. Example. Solve : 2y + 42:-2tf 10|i/-5fx-18 3a:+.y 13?/-37^_ 9-9a;-t/ 10a:+.25i/-10.5 L~T2^+ 44 -^^+ 22~^~ 33 ^^^ Process. From (1), 127 y+ 59 x= 1928 (3) From (2), 59 ^ + 127 rr = 1792 (4) Adding (3) and (4), 186 y + 186 a; = 3720 (5) Dividing (5) by 186, y+ a; = 20 (6) Subtracting (4) from (3), 68?/- 68 a; = 126 (7) Dividing (7) by 68, y - x = '2. (8) Adding (6) and (8), %y =22. .-.2^=11. Subtracting (8) from (6), 2x~ 18. .-. a:= 9. Solve SIMULTANEOUS SIMPLE EQUATIONS. 221 Exercise 84. 1 fy(^ + 7) = a:(y+ 1). (2y + Ax =1.2. I 2y+20 = 32;+ 1. XsAy -.02x= .01. r(y+l)(2:+2)-(y + 2)(2:+l) = -l. ^- \ 3 (y + 3) - 4 (2: + 4j = - 8. f .3 2: + .125y = a:- G. rx-4y = -3. l3a:-.5y = 28~.25y. I aj + v =32. .5y = 28~.25y. {x + y 6. -^ '4:X + Sy 2 y -\-7--x _ X--S To 24 -^"^ 5 9a;+52/ — 8 a; + y _ 7y 4- 6 12 4 9. ^ <^ 5 + y 12 + a; l2a;+ 53^ = 35. 10. -^ 3y-10(a:~l ) ^j-y . . _ ^ 6 -^ 4 "^ ^ - "• ' 4a;-3y-7 _ 3_^ _ 2 .y _ 5 5 " 10 15 6 * y~l , ? _ 3y _ 7/^^ 4. ? 4. JL L 3 "^2 20 15 "^ 6 ■*■ 10 -3 5 ~ 4 * U"^il 33 222 ELEMENTS OF ALGEBRA. r:. + l(3^.-y-l)^i+f(2/-l). f2x _Sy — 2 _ _ 4: + X y — x 14. ^ 18 by \2x. 2Zy -X 2x-^ = a? + 43 X — y 2.4 X + .32 y \ X -]r "i^o X — y .36 X - .05 \'. 1-3.V 7 = 2i. 3^/ + 11 -^-9 = = — ii;. « ^ _L 2.6 + .005 3/ 17. .5 ' .25 04 2/ + .1 .07 a? -.1 18. .6 _ Zx-2-\-,y ^ IBy + jx ^ 11 "^33 2x-h3y x-5 11 :?/ + 152 3 a; + 1 r ?/ - 2 10 - y ^ - 10 19 ^ ^ ^ ^ ■^22; + 4 0^ + 4?/ +12 I 8 21 H(2^+ 72/)- 1=1(20.-62/+!). I a? =: 4 2/. SIMULTANEOUS SIMPLE EQUATIONS. 223 Suggestion. Multiply the members of the first equation by 2, transpose, and unite like teims ; then clear the resulting equation of fractions. Multiply the second equation by 3, transpose, and unite like terms; etc. 23. < 24. < 25. r2 , y ^ 3?/ 1 ,1-1 + 2 = 1-2.. 6. f 6x + 9 3x-5 _ 3x + 4: 4 +4y~6~ i"^ 2 8a; 4- 7 Sx- 6y _ 9-4a ; ^10 2a:-8~ 5 16 + 60x _ 16xy- 107 3y-l ~ 5 + 2y • Suggestion. Multiply the members of the first equation by 5 + 2 y, transpose, and unite like terms ; then clear of fractions ; etc. 26. 27. X — y __1 X -\- y 5 * 13 3 y+2a;+3 4?/ — 5a;+6 3 ^ 19 6y-5a; + 4~ 3y + 2a;+ l' ry-x = l. r5(y+3) = 3(ic~2) + 2. 28. I y+ 1 _ y-1 _ 6 29. ^ 2 ^ 3 U— 1 X ~ 7' i^y+3~a: — 2* 224 ELEMENTS OF ALGEBRA. 30. 31. 32. < i(2 2/ + 7^) -1 = 1(2 2/- 6^+1). U = y 6y2-24a^+130 2y-4:x+ 3 151 - 16y _ 9 0^3/ -110 4^-1 3a;-4 ' ^4a; + 22/ 4a;+53/ ""16 31 "" • 2^+j/ 3 ?/ - 2 0? 36 ~5 +""~6 =y r5.T + 202/ = .l. 04 J 2/-^ 3-"- ^''- \ll^ + 302/ = -.9. l^+_^±i_7^Q V2/ — a; — 1 35. r 2a;— .5y 5jy— 19a;— 15 „ ^— a; + 2 93. Fractional simultaneous equations in which the unknown numbers occur in the denominators as simple or like expressions^ are readily solved without previously clearing of fractions. Thus, Example 1. Solve: ^'h 21 y = 10 20 6 X y - 2 (1) (2) Solution. 10 3 Dividing the members of (2) by 2, we have — — - = 1 (3). Mul- 70 21 tiplying the members of (3) by 7, — — -— = 7 (4). Adding the X y SIMULTANEOUS SIMPLE EQUATIONS. 225 members of (1) and the corresponding members of (4), we have . X = 5. Substituting this 85 5 — = 17. Dividing by 17, - = 1 X X value of X in (1), gives - = 1: .. y = 3. Note- If we cleared these equations of fractions they would give the pro- • liict xy, and thus become quite complex. In the solution of this particular .lass of examples it is always easier to eliminate one of the xmknown numbers without clearing of fractions. Example 2. Solve: 2-y^rx = ^ 2 4 _ 2 20 136 Process. Multiply (0 by i, 3T + g^- = ^ Subtract (3) from (2), Simplify (4), y 4_ 5x 2,1 x 20 ■ 27 X 8 135 X 208 9 208 9 • 2 2 Substitute in (2), ^ - 312 = -8, or « = 304. ^y ^y (1) (2) (3) (4) ^~ 390' y = 456 Exercise 86. Solve: 1. < 2. r2 1 ,^ - + - = 10. X y 9 y 2x~ ^' ^3y 4 a; 6 * 226 ELEMENTS OF ALGEBRA. 5_ 12 24 12. ?-^ = 16. 2/ 2a; 14,^ + - = _ 15. l2y X 7. y + l __7^ ~ 12' 12* _ 5 ~ 6' = 2. 13. 14. 2 2/ 4a? 13^/^22; 71 '1- 79. 15. < 5 16 - + — X y = 44. y^x y 9. ri5 8 17 2/ X ~ 3 2 3 7 I y a; 5 16. 11 2^ + 6 32/ 17. 17 6 1/ — 5 a; 3 2* 10. 1 + 2 (2; - 2) ' 3 (2 2/ 3 5 1) 5 a; -10 4 (4 2/ -2) = 5. = 1. 11. 22/ 2 17. 2a; 2 5 4 "27 1 42/ 1 11 "72 18. SIMULTANEOUS SIMPLE EQUATIONS. 227 a; — 2 y + 2 ^g 3 1 1 ^a; — 2 y +• 2 2 1 1 7 2x 3y""15* a: — 5 v/ + 4 a: y 1 4 a; V 15 19. Suggestion. Reduce the first member of the second equa- tion to mixed expressions. Etc. ('2i _ 3y ^ 2 rL2. 25^86 l3^ 3y-2a;+l 3'2lJ^2^ 5, 2y _ * |25_1_6^ a;"^3y-2a;+l la; y 94. In solving literal simultaneous equations, either of the pre- ceding methods of elimination may be applied, usually the method by addition or subtraction is to be preferred. Note. Numbers occupying like relations in the same problem, are generally represented by the same letter distinguished by different subscript figures ; as, «l ; «2 ; "8 > fitc* ? r«*d a one ; a ttoo ; a three ; etc. They may also be represented by different euxents ; as, a'; a"; a'"; etc.; read a prime; a second; a third; etc. Kt AMPLE 1. Solve: 1^ ^ + n y = a (1) -hn, iV = «i (2) Process. Multiply (1) by mj. m^mx + m^ny = m^a (3) Multiply (2) by m. m^mx + mn^y = ma (4) Subtract (4) from (3), m^ny — mn^y = mjO - -w»«i, or factoring, (m^n — mn^)y — m^a - - may Dividing by mjn — muj, ^""h - Oj m ^ nij^n - -mnj Multiply (1) by rij. mn^x -f- nn^y = n^a (6) Multiply (2) by n, m^ux -\- nn^y = na^ (6) Subtract (6) from (5), mn^x — m^ux = 71^0 - -no,. or factoring, (mui — in^n)x = n^a - - noj. Therefore, -_a^n -m^n' 228 ELEMENTS OF ALGEBRA. Example 2. Solve: ^ f X y 1^+5 + ^='" x + (1) (2) [ 2a6 ~ a2 + 62 Proceas. Free (2) from fractions, transpose, and factor, {a-hyx-{a^-hyy^Q (3) Simplify (1), {a-h)x^{a^lS)y = 2a(a+b) (a-b) (4) Multiply (4) by a -ft, (a-byx+(a^-b'^)y = 2a(a+b) {a-by (5) Subtract (3) from (5), 2 a (a + 6) i/ = 2 a (a + 6) (a - &)2. Divide by 2 a (a + 6), y = (a-b)\ Substitute in (1), Examples. Solve a+b + a-b = 2a. .'.x=(a + by. f m n—'m(m+n)(b—y)__ J n{a-\-x) m(b-~y) ~ ! ryt I — — + T = m2+n2 \^a + x b—y Process. From (1), Multiply (3) by ~, + n(a + x) m{b — y) = m + n + a + X m^(b — y) m — (m4-n) (2) (3) (4) Subtract (4) from (2), ^3^ - ^2Q,_y^ 1 Simplifying, Substitute m{b — y) ^ m m(b-y) = 1 or b-y n a + x m2 in (2), = n^ .-. X Example 4. Solve: < x-y +\ x-y-1 x + y + I x + y - a = -6 = (1) (2) SIMULTANEOUS SIMPLE EQUATIONS. 229 Procesa. From (1), {a- l)x - (a- l)y = a + I (3) From (2), (6- 1):. + (6- l)y = 6 + 1 (4) Divide (3) by a - 1, x~y = ^—^^ (5) Divide (4) by 6 - 1, x + y = ^-3-j (6) 2(a6-l) Add (5) and (6), 2 x = (q_i)(fc_i^ ' ab-l •••^-(a-l)(6-l)- 2(a-6) Subtract (5) from (6), 2 y = ^^. ^^^^.^^ • a-b Exercise 86. Solve : - (ax-\-hy = m. ^ ( ax -\- by = a^, '\bx-\-ay = n. ' \hx -\- ay = 1?. ^ nx + my = n. ^ (px'-qy = r, ' \px-\-qy = r, ' \rx—py = q, ^ (ax = hy. ^(x + ay = ai. ' \bx-\-ay = c, ' \ax-\-aiy=l. -? + ?=!. (^ + ^- = a. . J a b ab ^J^ \ m n I bi aibi ^n ' m 5. < 3j/^2^^2 r_y X ^ 1 m ?i * -^ Ja-{-b a — b a + b 9y_6^^3 ' I ?/ , ^ ^ ^ 230 ELEMENTS OF ALGEBRA. = (a + h) y. ' \ cy + hx = a. ' \x + y = c. rih; / J 14. - + 2' = 2. l-^ + r = cll+-). {mx = ny \^(m — n) y = {m -{• n) X. y X ^ a ai (ax-hy \(a-h)x+ (a + b)y=2 {a^ - l^). 22 (m{m-7j) = n(x + y-m). \m (x — n — y) = n (x — n). a b a fx + y+l^m + l 2g^a + a? b-y b 24. <^2/-^+l ^"1 b a b \ X + y + 1 __ n+1 a + x b — y~a \y — x-^l~l—n 25 / 3/ - ^ + 2 (m - ?i) = 0. ■ 1 (a:; + 7i) (y + m) — (y — m) (x — n) = 2 (m — n)\ SIMULTANEOUS SIMPLE EQUATIONS. 231 95. Simultaneous equations with three or more unknown num- bers are solved by eliminating one of the unknown numbers from the given equations ; then a second from the resulting equations ; and so on, until finally there is but one equation with one unknown niunber. Thus, r 2 y + 2 + 2 y = - 23 (1) y-f-32 = - 2 (2) 4a: + z=13 (3) Example 1. Solve 3 + 3. Process. Multiply (2) by 2, Subtract (5) from (1), Multiply (4) by 12, Subtract (7) from (3), Multiply (8) by 5, Add (9) and (6), Substitute in (4), Substitute in (3), Substitute in (2), - 20 (4) 2y + 6z = - 4 (5) -52 + 2y = - 19 (6) 4a; + 361; = -240 (7) z-36v= 253 (8) 52- 180i;= 1265 (9) -178y= 1246. .-. r = - 7. I -21 =-20. .-. x = 2. 12 + 2=13. .-. 2=1. y + 3 = -2. .-. y = -5. Proof. Substituting — 7 for y, 3 for x f - 23 = - - 2 = - 13=13 - 20 = - : (I), (2), (3), and (4), we have -{ — 5 for y, and 1 for z in 23 (1), ^ ^^]' identities. (3). (4), Kote. When the values of several unknown numbers are to be found, it is necessary to have as many simultaneous equations as there are unknown numbers. EiLAMPLE 2. Solve: J_ J 1__ 1 2z"^ 4y 32~ 4 1 1 4 0) (2) (3) \2 ELEMENTS OF ALGEBRA. Process. Multiply (1) by 2, - -f 1 2 1 2y 3z-2 (4) Subtract (2) from (4), 5 2 1 6y~ 3z~2 (5) Subtract (2) from (3), 2 4 I5y'^ z~^^ (6) Multiply (5) by 6, 5 4 (7) Add (6) and (7), 77 77 16 y~ 15- .'. y = :1. Substitute in (2), 1 1 .'. X- :3. Substitute in (5), fl 1 5 2 1 6 3z~ 2' .'. z - :2. (1) Example 3. Solve : - 1 1 ly + i-" Process. s (2) (3) Add (1), (2), and (3), ?+?+^ = a + & + c (4) Divide (4) by 2, ^ + J + 1 = ^±A±f (5) Subtract (3) from (5), ^ = ^±|-— Subtract (2) from (5), ^ « + c - 6 Subtract (1) from (5), y 2 1 &4-C — a 2 «+&-c 2 y — a-6 + c 2 h-\rC — a SIMULTANEOUS SIMPLE EQUATIONS. 233 Exercise 87 Solve: f'^x- y+ z= 9. I. ^ a:_2y+32= 14. r4:x-3y+2z = 40. ^<5x-i-9y-7z = 4:7. Ua;+8y-32 = 97. r2x-32j+oz= 15. 3. < 32:+27/- z= 8. V— a;+ 5y + 2s = 21. rSx-Sy+ z= 0. 4.<2a;-7y + 42:= 0. v9 2:+5 7/+32= 28. rx -\- y -\- z= 5. 6. ^ 3 7/-5x + 72 = 75. 19 y- 11 2+ 10= 0. r.65//- .95a: = .5. 6. < 5.1 a: -3.3 3 = 6. V20.3 2- 23.1a: = 21. rax •\- hy '\- cz=iZ. 7. < rt x — 6 y + c 2 = 1. Vaa: + 6y — • C2 = 1. r.2a; + .ly + .32=14. 8- < .52:+.4y+.a2 = 32. ^.7y-.8a:+.9c= 18. 9. < 2^ + 2 + 3 = -+2+1= a; ?/ X y z ^ + - + ' 6. --1, 17. = 1. = 1. = 1. 11. < 234 ELEMENTS OF ALGEBRA. fv-hx + y + z = 14:. \2v + x = 2y + z-2. 14^^ ^ Sv - X + 2 y + 2 z = 19. \ V X y z ^a — X h — y c — z + + = 0. X y 15. < a — X h c _ X y z~ X y z 0. 15. Suggestion. Reduce fractions to mixed expressions. Etc. 'x-\-2y = 9, 16. ^3 7/ + 4^ =14. 72 + V = 5. ^2v + 52:== 8. rx + y= 1. \x -\- z = b. 2 1_ 3 X y z 18. < ^ - - = 2. z y 1 1_4 x^ z~ ^' f4:y-\-Sx + z 2x + 2z-y-\-l _ y-z-5 10 15 19. < 9?/ + 52:-2;s 2y + a; — 3^_ 7x + z+S 1 r2 ^ 4 ~ 11 ^ 6 * 5^ + 32 2y+Sx-z ^ ^ Sy + 2x+7 —4 12 + 2. = 0.-1+ g Queries. Upon what principle is elimination by addition and subtraction performed ? What substitution ? What comparison ? SIMULTANEOUS SIMPLE EQUATIONS. 235 Miscellaneous Exercise 88. Solve : 'x+1 x-l 6 r4a;+y = ll. 23 y a(x^-y) + h{x-y) = l. & (x + y) = 1. 7. 4. 5. 10. X — a = 0. a b 0. 'a; + 2y=2-32-4i;. Sy + 2x = S — 4:z-5v. 9v-82-3 = -6a;--7y. ^v = 25 -4^- 16y- 64a:. ■(m2 — n2) (5 ^^ 4. 3 y) = (4 ^ _ n) 2 m n. „ a m vF m^y 3 a; 15 /M- lf-!=»- Ihh' y ^ 12-3- .|!+l-^ 1=^ ,ai bi ' -{-{m-\-n-\'a)nx = n^y-\-{m-\-2n)mn. 11. 3 V + a: + 2 y - 2 = 22. 4a;- y4-32 = 35. 4v + 3a;-2y = 19. .21^ + 4^+2^ = 46. (-15 a; = 24 2- 10 y + 41. 12. \ 15y=12a;- I62+ 10. ll8y-(7 2-13)-=14a;. 'Ul=z. 13. < X 1 a; ■ z y 2 + - = 11. ^U-3. y 2 236 ELEMENTS OF ALGEBRA. 14. 'x + y + z + v = 14:. 2x+Sy + 4:Z+5v = 54:. 4:x — 5y — 7z+9v = 10. :Sx + 4:y + 2z-3v = ll. 16. ) X + Z + V = = 5. 10. X + y+ V = 6. x + y + z=12. 15. 18. ax + by = 2m. ax -\- cz = 2 71. ,h y + c z = 6p, rmx + ny + pz = m. 17. < mx — ny—pz = n, \mx +py + nz = p. Sx 2y , \ly — + 1+ ^ 10 45 4.X-2 8 7 55^+ 71 y + 1 18 4 a; — 3 V + 5 45 ■ - + — 7 ^lx = v^2Z 17 + 2z-Stc. 20. 2(0 + 22/). 4. y= 2.25-\-.75u—5v. z = ll-^u. 19. {u=ly^ I X ( ax + \ay + b X by cy = m. cx = n. X y m 21. < -+ - = -. \x z jp I 1 1_ 1 \^z y n 22. f :^+2/+2; = a+J+c. i a+x = b-\-y = c+z. 9S 11-7^ 2(5-lly) ^ 17.5 + 5y 312.5-360a; ?j-x'^ ll{y-l)~ 3-2/ 36(a; + 5) ~ ^3 4 1 ^^ fx ^ ^ --^— + -= 7.6. -+l=4a;. X by z 4 _ .,- + - = 16.1. ^o X 2y z + l = 2y. SIMULTANEOUS SIMPLE EQUATIONS. 237 1 y + 1 y~ 26. < a; — X — i-i y l-x = 0. x-\-y = 2m^ 27. 711 -{-li- mn m+n m—n-\- mn m—n 28. m n V ^ - + - + - = 3. X y z m n P _ ^ X y"^ z~ ' 2 m X n 0. 29. 30. ^ xy x-{- y xz = 70. = 84. X ■{- z -^ = 140. y + z 31.^ xy x+ y yz y + z xz ^x -\- z xy — ^^ = m. 32. ^ 'ax-\- by + cz = 0. a^x + ly^y + (^z = 0. .a^x + h^y'i-c?^z = 0. 'a^z=:2. x + y xz x+ z yz _ ^y + « n. {ax + a^y — a^yVc 2(3a;-2y) _^_ bz-y 34 35. a; — 2g 3a; — 2i m — n + Sz-7 n 2x — Sz m ly-^ = w m + 71 2 7l6 7/1^ + 7?l 71 + 71^ 238 ELEMENTS OF ALGEBRA. CHAPTER XVII. PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS. 96. The solutions of the following problems lead to simultaneous simple equations of two or more unknown numbers. In the solution of such problems the conditions must be sufficient to give just as many equations as there are unknown numbers to be determined. Exercise 89. 1. If 5 be added to both numerator and denominator of a fraction, its value is f ; and if 3 be subtracted from both numerator and denominator, its value is ^. Find the fraction. Suggestion. Let x = the numerator, and y — the denominator. ra; + 5_ 3 — T~K — T' By the conditions, •{ y ^ I a:-3 _ 1 Solving these equations, x — 1, y =\\. Therefore, the fraction is ^y- 2. A certain fraction becomes equal to 3 when 9 is added to its numerator, and equal to 2 when 2 is sub- tracted from its denominator. Find the fraction. 3. Find two fractions with numerators 5 and 3, respec- tively, whose sum is ||, and if their denominators are interchanged their sum is f. PROBLEMS. 239 4. A certain fraction becomes equal to § when the denominator is increased by 3, and equal to J when the numerator is diminished by 3. Find the fraction. 5. A fraction which is equal to | is increased to {| when a certain number is added to both its numerator and de- nominator, and is \ when 3 more than the same number is subtracted from each. Find the fraction. 6. If a be added to the numerator of a certain fraction, its value is a ; and if a be added to its denominator, its value is ^ (a — 1). Find the fraction. 7. Find two numbers, such that two times the greater added to one fifth the less is 36 ; three times the greater subtracted from eight times the less, and the remainder divided by 9, the quotient is 7|. 8. Find two numbers, such that if the first be increased by a, it will be m times the second, and if the second be increased by 6, it will be n times the first. 9. Find two numbers, such that if to J of the sum you add 18, the result will be 21 ; and if from | their differ- ence you subtract |, the remainder is 3.65. 10. A farmer sold to one person 25 bushels of corn and 52 bushels of oats for S 38.30 ; to another person 42 bush- els of com, and 37 bushels of oats for $35.80. Find the number of dollars per bushel received for each. 11. A farmer sold a bushels of corn and b bushels of oats for m dollars ; also at the same time, c bushels of com and d bushels of oats for n dollars. Find the number of dollars per bushel received. Apply the result to 10. 240 ELEMENTS OF ALGEBRA. 12. A grocer bought a certain number of eggs, part at 2 for 3 cents and the rest at 5 for 8 cents, paying $7.50 for the whole. He sold them at 23f cents a dozen, and made $2 by the transaction. How many of each kind did he buy ? 13. A grocer bought a certain number of eggs, part at the rate of a eggs for m cents and the rest at the rate of h eggs for n cents, and paid c dollars for the whole. He sold them at d cents a dozen, and made ^ dollars by the transaction. How many of each kind did he buy ? Apply the result to 12. 14. A number is expressed by three digits. The sum of the digits is 8 ; the sum of the first and second exceeds the third by 4; and if 99 be added to the number, the digit in the units' and hundreds' place will be inter- changed, rind the numbers. Suggestion. Let 2 = the digit in units' place, and y = the digit in tens' place, also X — the digit in hundreds' place. Hence, 100a:+ 10y + 2 = the number, and 100 2+ 10^ + a: = the number with the digit in units' and hundreds' place interchanged. By the conditions, X + 2^ + z = 8, a: + 1/ — 4 = 2, 100 a: + 10 y + 2 + 99 = 100 z + 10 2/ + a:. Solving these equations, z — % y = 5, x — 1. Therefore, the number is 152. Note 1. In verifying, the results should be tested directly by the conditions of the problem. Thus, in the above, the sum of 2, 5, and 1 is, as one condition requires, 8. The sum of 1 and 5 exceeds 2 by 4» The sum of 162 and 99 is 251 fiB required. PROBLEMS. 241 15. A number is expressed by three digits. The middle digit is twice the left hand digit, and one less than the right hand digit. If 297 be added to the number, the order of tlie digits will be reversed. Find the number. 16. A number is expressed by three digits. The sum of the digits is 18 ; the number is equal to 99 times the sum of the first and third digits, and if 693 be subtracted from the number, the digit in the units' and hundreds' place will be interchanged. Find the number. 17. The sum of the three digits of a number is n ; the number is equal to a times the sum of the first and third digits, and if m be subtracted from the number, the digit in the \mits' and hundreds' place will be interchanged. Find the number. 18. If a certain number be divided by the sum of its two digits the quotient is 3, and the remainder 3 ; if the digits be interchanged, and the resulting number be di- vided by the sum of the digits, the quotient is 7, and the remainder 9. Find the number. 19. If a certain number be divided by the sum of its two digits the quotient is a, and the remainder b ; if the digits be interchanged, and the resulting number be di- vided by the sum of the digits, the quotient is c, and the remainder m. Find the number. 20. The sum of the three digits of a number is 16. If the number be divided by the sum of its hundreds' and units' digits the quotient is 77 and the remainder 6 ; and if it be divided hy the number expressed by its two right- hand digits, the quotient is 16 and the remainder 5. Find the number. 242 ELEMENTS OF ALGEBRA. 21. The sum of the three digits of a number is 9. If the number be divided by the difference of its hundreds' and units' digits, the quotient is 157, and the remainder 1; and if it be divided by the number expressed by its two right-hand digits, the quotient is 21. Find the number. 22. A, B, and C can together do a piece of work in 12 days ; A and B can together do it in 20 days ; B and C can together do it in 15 days. Find the time in which each can do the work. Suggestion. Let x = the number of days in. which A can do it, and y = the number of days in which B can do it, also 2 = the number of days in which C can do it. 1111111 111 Theequationsare- + ^ + ~ = 12' ^ + 2^= 20' ^^^ y + z = \b'^ from which a; = 60 and y = 2 = 30. 23. A and B can do a piece of work together in 48 days ; A and C in 30 days ; B and C in 26| days. How many days will it take each, and how many altogether, to doit? 24. A and B can do a piece of work together in a days; but if A had worked m times as fast, and B n times as fast, they would have finished it in c days. How many days will it take each to do it ? 25. A drawer will hold 24 arithmetics and 20 algebras; 6 arithmetics and 14 algebras will fill half of it. How many of each will it hold ? 26. A purse holds 19 crowns and 6 guineas ; 4 crowns and 5 guineas fill JJ of it. How many will it hold of each ? PROBLEMS. 243 27. A purse holds c crowns and a guineas ; ci crowns tn and ai guineas will fill — th of it. How many will it bold of each ? 28. A and B together could have completed a piece of work in 15 days, but after laboring together 6 days, A was left to finish it alone, which he did in 30 days. In how many days could each have performed the work alone ? 29. Two persons, A and B, could finish a piece of work in m days; they worked together a days when B was called off and A finished it in n days. In how many days could each do it ? 30. A can row 8 miles in 40 minutes down stream, and 14 miles in 1 hour and 45 minutes against the stream. Find the number of miles per hour that the stream flows, also that A rows in still water. Suggestion. Let x = the number of miles per hour that A can row in still water, and y = the number of miles per hour that the stream flows. Then, x + y = the number of miles per hour that A can row down the stream, and X — y = the number of miles per hour that he can row up the stream. Since the distance divided by the rate will give the time, by the conditions, 8 2 x + y~ 3* 31. A can row m miles in h hours down stream, and mi miles in ^i hours against the stream. Find the number of miles per hour that the stream flows, also that A rows in still water. Apply the result to problem 30. 244 ELEMENTS OF ALGEBRA. 32. A boatman sculls down a stream, which runs at the rate of 5 miles an hour, for a certain distance in 3 hours, and finds that it takes him 13 hours to return. Find the distance sculled down stream, and his rate of rowing in still water. 33. A man who can row at the rate of 15 miles an hour down stream, finds that it takes 3 times as long to come up the stream as to go down. Find the number of miles per hour that the stream flows. 34. A waterman rows 30 miles and back in 12 hours ; and he finds that he can row 3 miles against the stream in the same time as 5 miles with it. Fiud the number of hours in going and coming respectively ; also, the number of miles per hour of the stream. 35. A waterman can row down stream a distance of m miles and back again in h hours ; and he finds that he can row h miles against the stream in the same time he rows a miles with it. Find the number of hours in going and coming, respectively ; also the number of miles per hour of the stream, and his rate of rowing in still water. 36. Five pounds of sugar and 3 pounds of tea cost $2.05, but if the price of sugar was to rise 40 %, and the price of tea 20 % they would cost $2.51. Find the num- ber of cents in the cost of a pound of each. 37. If / pounds of sugar and h pounds of tea cost m dollars, and the price of sugar was to rise a % , and the price of tea h %, they would cost n dollars. Find the num- ber of cents in the cost of a pound of each. PROBLEMS. 245 38. The amount of a sum of money, at simple interest, for 11 months is S1055; and for 17 months it is S1085. Find the sum and the rate per cent of interest. 39. The amount of a sum of money, at simple interest, for VI months is a dollars ; and for n months it is h dollars. Find the sum and the rate of interest. 40. A grocer mixes three kinds of coffee. He can sell a mixture containing 2 pounds of the first kind, 9 pounds of the second, and 5 pounds of the third, at 18 cents per pound ; or one composed of 6 pounds of the first, 6 pounds of the second, and 9 pounds of the third, at 19 cents per pound ; or one composed of 5 pounds of tlie firet kind, 2 pounds of the second, and 18 pounds of the third, at 22 cents per pound. Find the number of cents in the cost of a pound of each kind. 41. The fore-wheel of a carriage makes 6 revolutions more than the hind-wheel in going 120 yards ; if the cir- cumference of the foie wheel be increased by J of its pres- ent size, and the circumference of the hind-wheel by J of its present size, the will be changed to 4. Find the number of yards in the circumference of each wheel. 42. The fore-wheel of a carriage makes a revolutions more than the hiud-wheel in going b feet. If the circum- ference of the fore-wheel be inci-eased by — th of itself, and 8 ^ that of the hind-wheel by - th of itself, the hind-wheel r will make c revolutions more than the fore-wheel. Find the circumference of each wheeL 246 ELEMENTS OF ALGEBRA. 43. A grocer has two kinds of coffee. He sells a pounds of the first kind, and h pounds of the second, for m dollars; or, ax pounds of the first kind, and hi pounds of the second, for mi dollars. Find the number of dollars in the price of a pound of each kind. 44. A jeweller has two silver cups, and for the two a single cover worth 90 cents. If he puts the cover upon the first cup it will be worth 1^ times as much as the other ; if he puts it upon the second cup it will be worth lyig times as much as the first. How many dollars in the value of each cup ? 45. A jeweller has two silver cups, and for the two a single cover worth a dollars. If he puts the cover upon the first cup, it will be worth m times as much as the other ; if he puts it upon the second cup it will be worth n times as much as the first. How many dollars in the value of each cup ? 46. A broker invests $5000 in 3's, $4000 in 4's, and has an income from both investments of $315.50. If his investment had been $1000 more in the 3's, and less in the 4's, his income would have been $5.50 greater. Find the market value of each class of bonds. Note 2. 3's means bonds which bear 3 % interest. The " quoted " price of a bond is its market value. Thus, a bond quoted at 115i means that a $100 bond can be bought for $115.50 in the market. 47. A broker invests m dollars in a's, n dollars in c's, and has an income from both investments of h dollars. If his investment had been d dollars less in the a's, and more in the c's, his income would have been p dollars less. Find the price paid for each kind of bonds. PROBLEMS. 247 « 48. A and B do a piece of work together in 30 days, for which they are to receive $1G0. But A is idle 8 days and B is idle 4 days, in consequence of which the work occupies 5J days more than it would otherwise have done. Find the number of dollars received by each. 49. A and B do a piece of work together in m days, for which they are to receive c dollars. But A is idle a days and B is idle h days, in consequence of which the work occupies n days more than it would otherwise have done. Find the number of dollars received by each. 50. The amount of a sum of money, at simple interest, for 5 years is S600; and for 8 years it is $660. Find the number of dollars in the sum, and the rate of interest. 51. The amount of a sum of money, at simple interest, for a years is m dollars ; and for h years it is n dollars. Find the number of dollars in the sum, and the rate of interest. 52. If a grocer sells a box of tea at 30 cts. a pound, he will make SI, but if he sells it at 22 cts. a pound, he will lose S3. Find the number of pounds in the box, and the number of cents in the cost of a pound. 53. The smaller of two numbers divided by the larger is .21, with a remainder .04162. The greater divided by the smaller is 4, with .742 for a remainder. Find the numbers. 54. The smaller of two numbers divided by the larger is a, with a remainder m. The greater divided by the smaller is h, with c ibr a remainder. Find the numbers. 248 ELEMENTS OF ALGEBRA. CHAPTEK XVIII. EXPONENTS. 97. An Exponent is a figure or term written at the right of and above a number or term (Art. 21). '^ m Thus, in the expressions 5^, a% 6", and (a + by; 2, c, —, and 3 are exponents. Zero Exponents. When the dividend and divisor are equal the quotient is 1. Thus, ^,= 1; -,= 1; ~,= i; ~ = I; etc. But (Art. 30), 32 = 32-2 = 30; ^ = aO; ^^ = «^ ^ = «^ etc. Therefore, it follows that a^ = I. Hence, in general, I. Any expression with zero for an exponent is 1. The Reciprocal of a number is unity divided by that number. Thus, the reciprocal of n is -; of n + m is n -\- m Negative Integral Exponents. a3 X a-8=a8-3 = «o^i Divide by a^, a» X a-« = «"-« = a^= \ Divide by a", a~** = — . Hence, in EXPONENTS. 249 II. A negative integral exponent indicates the reciprocal of the expression with a corrcspoiuling positive exponent. The expression a", where n is any positive integer, represents tht product o/n equal factors, each equal to a. It has been shown that : Art. 21, a^ X a* — a*»+". Art. 30, a"^ — a^ = a"*-", where m is greater than n. Art. 30, ti^ -T a'*= ^_,^ , where m is less than n. Art. 27, (a*")" = a*"", whatever the value of m. Thus, By Art. 21, a* X a" X a' x .. . . a"* = a»+«+ ''+••'». Take n factors of a*, a*, a**, a"*, and suppose each of the n ex- ponents equal to m, then it follows that (amy _ Qtnn^ Hencc, m can be positive or negative, i itegral or fractional. By II.. <'-" = ii- a"* Multiply by a"*, a-" X a"» = — • If m is greater than n, Art. 30, -;; = a*"-". Therefore, a-" >^ a*" = a"*-*. If m is less than n, a* 1 a» a i« — w By II., 7=^ = «' a" Therefore, a-» x a*" = «"•-'• for all possible integral values of m and n. 98. By Art. 27, (ai)" = a" X 6". Therefore, o-^*" = (aft)*. Similarly, a" X ft" X c" X . . . . p" = (a 6 c . . . . />)■ 250 ELEMENTS OF ALGEBRA. If n is a negative integer, «-" X *-" = a« X 6" =(«6)» =("*)-• Similarly, a-« X i-^ X c-» X ....jD-"= {ahc ... general. . Py- Hence, in I. The product of tv:o or more factors, each affected with the same exponent, is the same as their product affected wiih the exponent. By IL, Art. 97, a" -f &« = a« 6-». Also, a« 6-« = (a 6- 1)« = (? ) • Therefore, a** -f- ft" = [ r Similarly, a-" -^ &-»» = f r) Hence, in general, II. TAe quotient of any two factors, each affected with the same exponent, is the same as their quotient affected with the exponent. Illustrations: I. 22 X 3^ = (2 X 3)2 = (6)2 = 36 ; 28 X 3^ X 48 = (2 X 3 X 4)8 = (24)8 = 13824 ; 2-2« X 3-2« X 4-2" = (2 X 3 X 4)-2 « = [(24)2]-" ^ (576)-«= ^ ; (f)-2 X (f)-=^X (i)-2z. (| x f X i)-2 1 (})- = ^=16. 16\-8 II. 242-^62= (5^4)2= (4)2:^16; (_16)-8--(-4)-8=(--^) 1\4ot /'1\4'» 1 These examples are said to be simplified, that is, they are expressed in their simplest forms. EXPONENTS. 261 Exercise 90. Simplify : 1. (n^f X {a^f X (71)2; (J)2 X (2)2 X (§)2 - (if. 2. (a;* y-'")8 -f- (oj- V")^ (216 2^2)4^(54 2^-2)4^ 3. (|a)8 X (|aj-2)8; (^ri)-" X (^■)-; (a;)* x (2:"^)*. 4. (x)" X {a^Y; (f)-" X (})-" X (2)""; {a-Hf X (a6-8)6 5. (2 71)10 X (2- im)iO; (a 6-ic-2)3 - (a-ife-ac-*?/^)^. 6. (4a*^a:«)-"^(2-2a-3*2;-V)"'*; (^-iy*)-^-^(^^rV. 7. a--- X (3 2>"')-" X (ci)-™; (:r)i- - (^'y-". 8. («-2 5)-2 X (« ?>-3)-2; (rtS J8 + ^6)- 3 ^ (a6-an3)-8 9. (i)" X {^T X (J)" ; (a2'' + a" ?»2'')-i x («" - &2«)-i. 10. («-i)-^ X (xi)-5 X (x-t)-6 X (at) -5 X (&i)-^ 11. (^^"+*)'' X (i2-*j" X (an)" X {b-nY; aS -^ (2 a)8 12. « 2x(2a)-2-f.^^y; (2«-2)-2x^^y'x(|a)-2. 13. (a-i V^x)-' X (.r-2v^6)-8. (,,i)2'' X (2^)2" X (c")2". 14. (2")-" X (2"-!)-'' X (2-2—1)— X (2-2"+i>-'» X (r)-\ 15. (2''+i)"' X (2— "+")"• X (2"'-!)'" X (4-"-^)'" -T-(ir))-'". 16. [(2:-y)-8]-X[(rr + 7/)-]-8; (|)-" x ff)- - (J)- 252 ELEMENTS OP ALGEBRA. 99. Positive Fractional Exponents. If m and n are both positive integers, Kan) = a"*. m Take the nth root of both members, a'* = \/a^. m Therefore, a« means the nth root of the mth power of a, or the mth. power of the nth root of a. Hence, The numerator, in a fractional exponent, denotes a power, and the denominator a root. The denominator of the exponent corresponds to the index of the root. Thus, (81)1 = \/{Siy = (^8iy = (3)3 = 27. m In a« = /y/a"*, m is the index of the power, and n is the index of the root ; also a, m, and n may be any numbers. The expression may be raised to the power indicated by the numerator of the expo- nent and then extract the root of the result indicated by the denomi- nator; or, extract the root first and then raise the result to the power indicated by the numerator of the exponent. Thus, (-8)1 - -V^FS? = v'64 = 4 ; or, (- 8)f = (-v^^)' = {~ ^Y = 4- Notes: 1. a-« is read "a exponent —n;" a" is read "a exponent -; a~ « is read "a exponent ." These are abbreviated forms for "a with an exponent —n; etc. m 2. It is manifestly incorrect to read a« " the - th power of a." There is no such thing as a fractional power. 3. We must be careful to notice the difference between the signification of a fraction used as an exponent, and its common signification. Thus, f used as an exponent signifies that a number is resolved into five equal factors, and tlie product of four of them taken. mXc mc 100. By Art. 73, a»» = a« x " = anc; m e m 7H-T-C n also, a** = a" ^ " = a^ Hence, EXPONENTS. 263 L Multiplying or dividing the terms of a fractional ex- ponent hy the same number will not change the value of the expression. ^n^a^-"^. But a^^^=^a^, and ^a^=^/^a. Therefore, Jrn - \/;^a. Hence, in general, II. The mnth root of a number is equal to the mth root of the nth root of that number. niuBtrations. 2* = 2* ; 6» = 6i ; 62« = 6«^; ^^64 = '^ ^64 = ^ = 2. 101. Negative Fractional Exponents. If m and ti are both positive integers, ( --Y \a »•/ = a""*. By II., Art 97, a-« = ^. Take the nth root of both members, m 1 a » = — . Hence, Ajiy expression affected with a negative fractional expo- nent is equal to the rccipror/U of the expression with a cor- responding positive exponent. _"* 1 "• 1 Notes : 1 . From the relation a »• = -^ , a" = — ^ . Hence, the method of o* a * Art. 30 is true for fractional exponents. 2. Any factor of the dividend may be removed to the divisor (or from the numerator to the denominator of a fraction), or any factor of the divisor to the dividend, hy changing the sign of its exi>onent. 254 ELEMENTS OF ALGEBRA. Illustrations. 2^^ = ^ = ^^1 (1)"^ = (^ = | = I' ^ = «"^ ,,.-1 -3 1 1 1 2* X 3 3 / u^ X (i) ' X 4 ' = ^, X I X -3 = -j,~ = - t-i ^x-")~ 1 11:^, -^ -^ - x^ ' x~ x^-^ • 102. (ah^y=ab. Take the nth root of both members, J 1 1 1 j^ Similarly,* an x &" X c« X . . . . ;?" = (a 6 c . . . . ;?>. Hence, I%e product of two or more factor's each affected with the sa7ne root index, is the same as their product affected with the root index. In the same manner we can prove that Kotes : * 1. If we suppose that there are m factors oi a,h, c, p, and that each factor is equal to a, then it follows that By Art. 99, [a^Jn = an. Therefore, Va»/ = an. 2. Similarly, \an) = a. Hence, Tfie nth power of the nth root of a number is equal to that number. Illustrations. (A)* X (f,)^ X 8* = (f X | 8)* = ^'^ = |; EXPONENTS. 255 »v 103. (a" X a'T' = (j««+»*. m b JL Take the ncth root, a« X a« = (0™'+"*) m . 6 By Art. 99, (a"» «+"«>)"'' = a" ''^ «. m 6 m 6 Therefore, a»Xae = a«'^«. • iw ft r t ?j.*j.'"i. ? Similarly, a* X a« X a» X • • a« = a» « i"*""" «. Hence, I. Th£ product of several expressions consisting of the same factory affected with any exponent, is the factor with an exponent equal to the sum of the exponents of the factors. By Arts. 101, 21, c* -f- ac = a* X a « = a»» «. Hence, II. The quotient of two expressions coTisistiiig of the same factory affected with any exponent, is the factor with an exponent equal to that of tlic divideiid mimis tJiat of the lUuBtrationa : I. 5* X 5"^ X 5 = 5*"*"^* = 5* = >^125 ; X* X a:* X a:" = x"+*. II. 2*-r2* = 2^"* = 2i = v^2; (a + 6)' -^ (a+ 6)* = (« + 6)'"* = (0 + 6)- A. Exercise 91. Simplify : 1. 16-f X 16-i; 25-i X 25i; 3^ x (^; a"! X ^. n n 6_»» X 2 m 2. aixa^Xa^; n-^' X n '; m "Xw ''; 2-iV2. 3. y"^ X 7/" » ; a' -T- a~'; rt* X J; (a^)* -r- (a^)h 4. (-2)-i-(-32H; a* -at; (^-J - (..y)-i. 256 ELEMENTS OF ALGEBRA. 5. x^ ^ rr2«; a '^-^m ^ ; (a- h)^ - (ah + hi)i. 6. 32« -^ 3" ; (a - &)" ^(a-b); (x-^/2)h ^ (// 2;^)?. \a^b ^J \a-^b^J \ax-^ J \x ^ J 8. {a -2 xf x{a-2xf x{2x-afx{ci-2 x^. 10. (.r + yy-'' ^{x + ^)-"; a3^+2y _^ a2^-3^; lA-^rA. / m \ ^ / »« — 1 \ —1— / \ "' + ^" / \ ^ 11 1*^ Xmnp / 3? \mn/) f CC\ n f CC\m 13. at -=- ai; 2" x (2»)»-' x 2" + ' X 2»-i x 4-". ^^2 . i^y {off (x'f . , ; .. , ^ /^V''" X04. caa"'=(a")-=a-. Take the n qih. root of the first and last members, (m r ( \p mp rp tp The principles of this chapter are true, whatever the values of a,h,c,....m, n, p, and q ; that is, a,b,c, m, n, p, and q can be positive or negative, integral or fractional. EXPONENTS. 257 niustrations. (2' X 3* X 4"^)' = s'""* X S*""' X 4-*'** = 2« X 2* X 3* X 4-1 = ^2 X V3 ; • L(a"^)"«]fl^ -r- I [(aW^JJlJ N m a"« »• = a-* — a = a. 105. Negative and Fractional Root Indices. _ j>i_ _w \ 1 an V« m _m _ e _£ £ 1 1 Similarly, y^a™ = a»« = a « = — = "^TZ* Hence, ^ negative root index, either integral or fractional^ indi- cates iJie reciprocal of Uie expression with a corresponding positive index. Note. Since it is impossible to extract a fractional or negative root, or raise an expression to a fractional or negative power, in order to perform the opera- tion indicated by such indices some preliminary transformations must be made. lUuBtrationB. ~i/^ = -: — = -| = -r = —^ ; i/4a« = (4 ay _ I _ ± ^_ _ ± _ J- = 1 a* Exercise 92. Simplify : 1. 1^27; V^; |/32 m-iO; Vsla-^; V^. 2. 1^8; [(63)2(a*)8(6-8)(a-6J-i)2]6; ^8a*6— ic"-2 17 258 ELEMENTS OF ALGEBRA. 2 V ^ / \y'J Kni^n^J ' vo«'"^'*^"^' V25' 1 1^ / J_ \a2 - 62 Queries. What does a negative exponent indicate 1 A fractional exponent ? A negative fractional root index ? Any expression with for its exponent = ? Why ? What is the product of as and a^ 1 Prove it. Miscellaneous Exercise 93. Express with fractional exponents and negative power indices : 1. ~\^^; ~^; 4"^; 'V^; (^a)^; ^^5^2 m Express with radical signs and negative integral root indices : m O 3. a-t; a^hic~^; 4:ah~^; 7a~^x~''; — - . X 4 EXPONENTS. 259 Express with radical signs and fractional root indices : n n 9 1 m m X T7"« 4. at; (4a2)!; alz^U; a'"5"; a" ft"; xy. Express with fmctional exponents and fractional power indices : 5. ^Jbi; y/2'6-; 3v'(8a-8)-|; ^a'~^; Va; ^^5. Express in the form of integral expressions : Sa^b 5 m* a x~^ x~^ a^ c-2 ' ahc' n-r 4^1' 4^^' -^-|' aib-l if Express with literal factors transposed from the numera- tors to the denominators: Simplify and express with positive exponents : 8. 4^; v^^^a. yj(h^; "-^/^; «-»; [Va^ H- V^]"". _ 2 ai X 3 a-i a 2^ x a'^ x v"^ «/ ^ s.-y 9. -== ; —=r. — ^^ ; Vm-3 -^ V7/il 10. 2;-ix2a;-i; (^V*; a^ x «i X a-J. 11. -^4^; (j^y^; V^^h X ^J'^F-s 12. aUiaxa-i6-U-i; (f^)"'; y/^- 260 ELEMENTS OP ALGEBRA. 13. aH^c^ Xa-H-^c-i; \l^\ i/(^~H*)^. 14. y" X y X "; (x + y)^ X n^ X n~r^ X Vti. 15. Y/(m^'^V'; y/«»6"i i(|)"'; -tF^. 16. (77i-i\^a)-3 X 'v/(a-2 V^"; y^^' + — 17. VaF^W^'-^{ah~~^y; l^^ll^. 2"' 2»2 a; '■m "^ n ^^- m2 ^ m-w Ml , J' (m2-7^2)2 2"(2"-i)" 1 2" + ^ 4""^^ 2»+i X 2"-" 4^"' (22»)"-i • (2"-i)" + > (9"x32x5kV27» V / / ; (2" X S"*)"" Is" X 6*"")"' 23. pir^^ EXPONENTS. 261 Multiply : 24 a*fe~i - a^h"^* + 1 by a^h~^' + 1. 25. a^ + a2»&*' + h^' by a" — a^"^)*"* + ^'. 26. a^h '—a* 5 •+« *7>i— a »6« by a«6 i + a "ft*. Divide : 27. a^ + rt^ M"' + &• by a» + a2^ M"' + ft*"., 28. a,-*"*"-*) — y2m(m-l) ^y ^{n-l)^^(m-« 29. a;**"* - if^-* by ar"*"'' + i/^'"-^ 30. a;^"'-*" - ^m'-am by ^"^-^"^ ± f*-'*, 31. a^ —3^+4:a*'*x*'—4:a^x^' by a2* + 2a^2:<'' — .r^*. 32. a3+a~i* by a5 + rt-^; riT/i + mx^ by n^yi + w^a;^. Separate into two factors : 33. a-^-b; a'^ - ft-f ; aV - 6-2«. Expand : 34. (a-U-6-ia:)*; (a:-2a;-i)8; [(a"^ -«?)']". Resolve into prime factors, and find the products of: 35. N?'!^, 4^2, \/96, \^ 36. ^12, v^72, \^, ^, ^^^576, V2l 262 ELEMENTS OF ALGEBRA. Find the cube roots of : 38. 8 a-2 - 12 a- V- + 6 a-f - a"! 39. a;3__9^+27a:-i-27ar-3. Find the 6th roots of : 40. a;« + ^ - 6 (a;* + ^) + 15 (^x^ + i) - 20. 41. 729 - 2916 a2« + 4860 «*" - 4320 a^" + 2160 a^" -576^10" + 64a^2«, 42. a;- 12 - 6 :r- 10 + 1 5 a;- 8 - 20 ^^- 6 + 1 5 ic- 4 - 6 aj- 2 + 1. Simplify and express with positive exponent : !i+i, ifan i/4 X 4"-i H:i, y/4«-i >^ 4„ + i 4fi ^"^"^ [(8a-6?>)2"]5" Q 9(a;0 + 2/0 + ^)-2m3' [(4a-3 6)5"]2''' ^ + T • .,, (20a3H8a;2?/2-12y^)" (m^ + ^^)^ (^^ - n^)^ [4 (2:2 + 2/2).f ' ' m6--?i6 RADICAL EXPRESSIONS. 263 CHAPTEE XIX. RADICAL EXPRESSIONS. 106. A Surd is an indicated root that cannot be exactly- obtained ; as, V5 ; 'V^f ; \^a^. The Order of a surd is indicated by the root index. Surds are said to be of the second^ third y fourth, etc., or nth order, according as the second, third, fourth, etc., or nth roots are retjuired. Thus, 'v/a, ^a, \/b, etc., yx, are quadratic, cubic, biquadratic, etc. Surds are of the same order when they have the same root index ; as, ^b, ^a\ and ^¥. A surd is in its simplest form when the expression un- der the radical sign is integral, and in the lowest degree possible ; as, ^32 a* = \/2^ a^ x 4 a = 2 a v^4 a. Similar or Like Surds are those which, when reduced to their simplest forms, have the mme surd factor ; as, 3 \/3 and a/3 ; 2 a vh and c ^/h. Otherwise the surds are dissimilar. Hotes : 1 . When a surd is expressed by means of the radical sign, it is called a Badical ExpressioxL 2. An Irrational Expression is one which involves a surd ; as, V3 ; a -\-h \c^. 3. An indicated root may have the form of a surd, without really being a 8uixl. Tims, Vi and Va» have the f(rrm of surds. 4. Rational factors or expressions are those which are not surds ; as, 2; a*x — bf^y. 5. Since a" — a^P, surds of the form Va^ and fo^ are equivalent sards of different orders. 264 ELEMENTS OF ALGEBRA. 6. A Mixed Surd is the product of a rational factor and a surd factor ; as, a V6 ; 3 Vb. 7. An Entire Surd is one in which there is no rational factor outside of the radical sign; as, V2; \'a^; Vx. 8. A binomial surd has two terms, and involves one or two surds; as, a -\-b Vx] a Vx — b yy- A compound surd or polynomial has two or more 2 3 - 4 - terms, and involves one or more surds ; as, y2 + 3 4/4 — 5 V3 ; a-\-h- c + 2Va. 9. Quadratic surds are of most frequent occurrence. 107. The methods for operating with surds follow from an appli- cation of the principles of Chapter XVIII. Thus, f = V^f . 2 a2 &3 == ^(2^2y3y3 ^ ^^^;^9; j^ general, n a = a^ = a^— ^a^. Hence, I. To Reduce a Rational Factor to the Form of a Surd of any Order. Raise it to the power indicated by the root index, and place it under the radical sign. 2V'3 = V2' X 3 = Vl^. f ^9 r= ^(1)3 X 9 = ^|. In general, n \ 1 a ^x = an xn — (a'*;r)» = y'a^. Hence, II. To Change a Mixed Surd to the Form of an Entire Surd. Reduce the rational factor to the form of the surd, multiply by the surd factor, and place the product under the radical sign. V72 = V62 X 2 = 6 V2. ^1029 a* = (7^a^ X Says = 7 a ^3a. 9 3/7 _ 9 i3/iZi _ ?Jl1 ,3/T ^.VJI _ «,V 1 X^'« ^Vu-^y2x4~ 2 - V4. 2V2a3 - 2V2a3 X 23a a / Sa ^j — "^ 2 V 2^ = i V 8 a- In general , Hence, RADICAL EXPRESSIONS. 265 III. To Reduce a Snrd to its Simplest Form, ii tiie surd is integral, remove from under the radical sij^n all factors of which the indicated root can be exactly obtained. If the surd is fractional, multiply its numerator and denominator by such expression that the indicated root of the denominator can be exactly obtaiued. \/2^ a X v^o^ = \^'2^a x a^ = a \/2. In general, _ — — A J * * * ^a X ^/b X ^c X . . . . ^ = a'* X b" X C^ X . . . . p^ = (abc . . . . py = \^a be p. Hence, IV. To Find the Product of Two or More Surds of the Same Order. Take the product of the expressions under the radical signs* and retain the root index. In general, ^/^-^\/r=(^y=\/'f- Hence, V. To Find the Quotient of Two Surds of the Same Order. Take the quotient of the expressions under the radical signs and retain thn root index. f/i5^64 = ^64 = 2. (/ V25^ = 1^(2»)'' = 2^ = 4. In general, ^'^ = (""j* = a^ = "^a. Hence, VI. To Find the vith. Root of the lith Root of an Expres- sion. Take the mnth root of the expression. Note. It is sometimes easier to j>erfonn operations with .simls if the arith- metical numbers contained in the surds be expressed in their prime factorSf and fractional exponents be used instead of radical signs, 266 ELEMENTS OF ALGEBRA. Exercise 94. Express in the form of surds of the 3d and nth orders, respectively : 1. 1; |; 22; 4"; 2 a"; Sahc; S x; a^; of; af'y\ Express as entire surds : 2. JV2; 1^3; 5 V32 ; f V^; leVflf; abVbi. 3. a 4/d^ b 6-8 ; 3 a^ ^ofc^ ^ i '^^ ; 2x</J^; | ^. 5. 5-;.^^25^i; (,»-l)v/^; '^^±^\I^^EI . ' ^ m — 1 771 — 71^ m + n ^' ;r^V-^r^' ^V^?^' ^"V^r-; ^^Vs^' Express in their simplest forms : 8. -^288; 3V150; •^^^^IIS?; fV90|; 2 a^W^. 9. V3i; ^Jl|; ^J^; ^'1029; Vf; V^ ; ^|- 10. <1\ ;J|; ^=T08^a, ^3^i;;i5ro, ^Z'^- 11. ^^?n=^^\ ^'V^; !^v/— • RADICAL EXPRESSIONS. 267 . -^', ^7290a3-j6m^2. ^^J^; ^a^^+Y"- 12. ,/ ^586 13. V(a; + y) (a^ - ^) ; Va«2 - 8 aa; 4- 16 a. 14. ^^yJtlEI^LlIl, ^ 1715 ^^-V^ Simplify : 15. Vl2 X Vl8 X V24; V54 -- Vl) V^Vlf. 16. ^Fex ^^54x^/128; [v0[28^ -h ^5^6^] ^ >^9^. 17. ^v^^:^.- X V50a8 66^ V32a63 18. V2^ a8 ^6 X vOUe a2 m2 a:^ X v^56 a^ m^ x\ 19. (^53a«fe9 -4- -^25 a* 6^) x ^125 aH X <^W^. 20. (V6M -^ V63~?) X v^54^ -T- v/feT: \/'^^^^^. 21. (^iiT?^ X ^a-ift-ic) -^'(v^a:-^oyo x ^^lO^X 22. (^|^^)^V20736; 'i^ivF^ - aX^. 23. Vf a8 X Vf a-2 X V.f ai X V2.5 a"*. 24. \1\J </W^^^\ (16 aH2)i X (ai h^f -r- ^2 J h. 108. ^5/2 = •^^21^ =^8. 3^=3*^]|^2^M? = 3]!J^l6. In general, p pxw y^aP = «" (n > /)) = a* »< "• = "^aP"*. Hence, 268 ELEMENTS OF ALGEBRA. I. To Reduce a Surd, in its Simplest Form, to an Equivalent Surd of a Different Order. Divide the required root index by the root index of the surd, and multiply the power and root index by the quotient. TheL.C.M. oftheroot indices (3, 9, 6) is 18. In general, pm Pin ^6w = fe"*" (m > pi) =: "^6p.». Hence, II. To Reduce Surds, in their Simplest Forms, to Equiva- lent Surds of the Same Lowest Order. Divide the L. C. M. of the indices by each index in succession. Multiply the power and root index of the first surd by the first quotient, of the second surd by the second quotient, and so on. Exercise 96. Express as surds of the 12th order: 1. A^2; ^3; f^; 3^2; ^a^; ^1; i^^S. ^ 2. a/sS; 1^32; ^a^; v'^^X V^^^i"^. Express as surds of the 7ith order, with positive expo- nents : 3. ^x^; V^; ah; ^'^j}; -L; v/«~"; ^. RADICAL EXPRESSIONS. 269 Reduce the following to equivalent surds of the same lowest order: 4 V5, ^11, 4^; a/2, \^5, \/3; \^8, V3, ^6. 5. ^2, ^8, ^i; v^7, ^5, ^6; Va, ^a^ G. '^^, Va; ^^«, ^a6, ^a^ ^1^?, '^^^. 8. Vaic^, ^/a^Q^\ ^fm, "^n, v^, ^mnx. 9. v"^, \^6^ ^;?; 41^5^, 2^VlW^, 10 a a/37. 109. fV6=A/(IF^^=A/i =a/W. i a/5 = a/(|)'' X 5 = V¥ = a/H .-. IV'5>Ia/6- In general, _ J a J^x — (a" x)" = y^a* as, 5 ^'y = (6- y)* = .y^Fy. Hence, I. To Compare Surds of the Same Order. Reduce them to entire surds, and couipaie the resulting surd factors. \ ^52"= ^{\y X 2« X 13 = y ^' =^^/42:25, I ^8 = ^{\y X 2 = ^ (f)« X 22 = ^45.5625, 3 Vl = a/3* X f = '^3« X (fli» = >^46.656. Therefore, the order of magnitude is 3 ^\, \ ^, \ >^52. In general, — ??J? JL 6 y^y = 6»»» ir ■ = "v^b^-y*. Hence, 270 ELEMENTS OF ALGEBRA. II. To Compare Surds of Different Orders. Reduce them to entire surds of the same order, and compare the resulting surd factors. Exercise 96. Which is the greater ? 1. 3 V6 or 2 Vl4 ; 6 Vll or 5 VlSf ; 4 VG or 6 Vi 2. 10V5or4V3l; iVTorfVlO; ^^2 or ^3. 3. V|or^l|; ^4 or ^5; Vf or ^T|. 4. ^11 or '^f; 1.6 or J ylO; \^6^ or V^. Arrange in order of magnitude : 5. V3, </4, </7 ; 8 V2, 5 a/5, 4 V7|. 6. 2'^2l, 3^49, 4V7; 3^4, 4 ^I|, 2^131. 7. 3 '^2, 3V2, |A^4; 2^21, 3^8, 2V8. Show that the following are similar surds : 8. ViO, V90, Vf ; J V'20, i V45, 5 Vf. 9. 7 V|, 'V/ff , 3 VS ; -^162, 3 ^32, </2E92. 10. V27, Vr92, Vl47, Vl; a^W^^ h</W^^ f V— . 11 QiV*^ ^V5T2 ^K^l^- i/«^ .V«*^'^ ^.la^(?m^ RADICAL EXPRESSIONS. 271 110. Addition and Subtraction of Surds. iVH"^=-iV^|x2 = -fV2. Adding, 3»a«Xa^arx(26)g 3 a o/-—— 26X726? =26^^^ ^ ^' 3/27a«x s/ -°V 26- -y-2hxm = ~ 26 ^^^«'*'^' 1 8/46^ 13/4 62a: X a* 1 3/ — sttt- 6V^ = ^V^=^3r^^ =^lJ/4a262x. Adding, = ^^-y- ^4a^b^x. Hence, Reduce each surd to its simplest form. Prefix the sum or differ- ence of the rational factors to the common surd factor of the similar surds. Connect dissimilar surds by their signs. Exercise 97. Simplify : 1. 3 Vis - 2 \/20 + 3 V5 ; 3 \/| + 2 V^. 2. 2V| + 3Vi; 2^l62-J^^^; 3 V^ + ^?^. 272 ELEMENTS OF ALGEBRA. 3. A/3 + Vli-2V5i; 5v^^=^54-2v/::r6 + -i'^685. 5. S^J^+^Ml-W^; 3\/l62-7^!^32 + ^1250. • 6. ^?+ 1--^^- 3 V^27^2. ^40-3 v''320 + 4^^135. 7. V50ab^c^-VS2aH^- (4:hc^-3ac)V2al)c. 8. a; Vwi^^ 71^"^ x^ — m vm^ n^^ x^^ + n -^m'^ n^ aP. 9. V3 a Z^'^ + 6 a& + 3 a + a/3 a &2 _ (3 ^ ^ + 3 ^^ 10. y/^+Y/^_2« + «^-^V^^^. ^ a — ^ a -{- a^ — h^ 11- |^A + 0-SVf-iVV96+1.5^|-ii^T750 + 8V|. 111. Multiplication of Surds. 3 -v/a X 7 V^ = 21 y4 X 3 = 42 y'3. /^2 X ^3 - lJ/2^"xT3 = :^432. f V2 X 1^3 X lA^ X -^i = f X I X f ^£6 X 3^ X {\f X (i)« = <{^28= -^2. In general, a ^x X h y^y = a x"" X h y^ = ah {x ?/)« = a 6 -y^ari/. a yar X h 'J^y = ax"" X b y'^ = a &(»;'"?/")"'« =: a & y'a;"' i/**. Hence, RADICAL EXPRESSIONS. 273 I. To Find the Product of two or more Monomials. Reduce the surds to the same order (if necessary). Prefix the product of the rational factoi-s to the product of the surd factors. Multiplicand, Multiplier, 3V3, 2^6, 3V2- 3V3H 9^6- 9V«H -2>^5 h2^ 3 >y/2-2v^5 multiplied by 3/y/2-2'v^5 multiplied by Sum of partial products. Hence, -6^5<^X3» 6 ^2«X 62-4/^3(7 h 6 v^288 - 6 -^'675-4 v^30. II. To Find the Product of two Polynomials. Proceed as in Art. 24. (^^/2-\-2^3){^^2-2^/3) = (3 X 2*)^-(2 X 3*)^^ = 32 X 2-2^ X 3 = 6. (a^x-^b\/~y) (a^x-bx/y) = (a x^^ - (hy^f = a^x-h^y. Hence, III. The product of the sum and difference of two binomial quadratic surds is a rational expression. Exercise 98. Simplify : 1. 2'v/r^ X 3 V3; SVf X J\/T62; J VlO x J^ Vl2j. 2\/l4 X V2i; 3^1 X gVJ. (5 V3-5) X 2 V3; \!^64x2V2- 4. (V2 + V3 + 2 \/5) X V2 ; 4 ^75 X 2 V^. 5. J V4 X v^iO; i VJ X § V^l; V5 X ^2. 2. J ^4 X 3 v^2 3. 3 >^3 X 3 V2 6. 3 \/| X ^1 ; J \/§ X 9 -^1 X \^. 18 274 ELEMENTS OF ALGEBRA. 7. 2^3 X'v/2 X J'^i; V^%X</'i; ^T68 x '^147. 8. '^^X^9X^9*; (3V2-3'v/6-V8+3V'20)x3V2. 9. a/5 X V^IO ; (Vn - Vm) xVn; 4 \/^ x 3 VS. 10. Vmn X \^S m^x X V2 nx. 11. "^/m^no^ X "^m^ n x ; 2 Va X '^^ X 3 ^a x '^^. 12. ^^2^ X ^3^ X V -i-; ^^(4 7/^ a^'^)" X ^(:2m^x)\ 13. V-xtV^r^; (V2-3V3)(2V3 + 3V2). 71 ▼ 71 3 ▼ 2 ft* 14 (3V5-4^2)(2^5 + 3V2); (^2 + ^3)^. 15. (5 V3 - 6 a/2 + a/5) (lO a/3 + 12 a/2 - 2 a/5). 16. (V2 + </l+ '^D «/2 - V3); '^24 X 6 ^3. 17. (V2 + V3) (V2 - VS); (^3 + ^4) (^3 - m 18. (a/5-a/3)(V5 + a/3); (a/5 + 2 a/3) (a/5 - 2 a/3). 19. y^l2 + a/19 X y^l2- a/19; v'TG X a/S. 20. y'9 + A/n X \^9 - A/rZ ; a/3 X ^2 X ^|. 21. (^a3 + ^^) (^-2 _ ^-3) . y| ^ ^1 RADICAL EXPRESSIONS. 275 22. V^10+ V68 X y^lO- V68; -v^^^xy^^. 23. {dVx+3 Va-^x) (5 Vi - 3 Va-2^x), 24. ^S X ^M; VW X ^i; {mfi<^Mf' 112. To Rationalize Surd Denominators of Fractions. 2 2 X V3 2 V3 2 ;^ = 7^^ -7- = -^ = T^ X 1.732 + = .23 + . 2 2X -v/3^« 2 ^3« 2 ,/_ — =r= = - — !^, = — - — (n>m) = — T . Hence, I. If the Fraction be of the Form — — = . Multiply both terms by y"^^^^- ^ ^^ 3+ V5 _ (3 + ys) X (3 + ys) _ 3« + 2 (3) ( /y/s) + (^5) * 3 - V5 ~ (3 - V5) X (3 + Vs) ~ 32 - ( VS)^ 14 + «V5 7 + 3X2.236+ ^ ^^ = 9-5 = 2 = ^•®^^"^- 4V3 + 3a/5 (4 a/3 + 3^5) X (2V7-3 V2) 2 V7 + 3 >v/2 ~ (2 V7 + 3 V2) X (2 a/7 - 3 v^ _ 8 \/2l + 6 a/35 - 12^/6 - 9 a/To g ox(A/^TA/g) _ aiVbTVc) _ a{\/bT\^c) V^±A/^"(A/^iA/^)x(A/ftTA/^)" W~-W~ ^"^ g a X Tfe T A/g) _ ajh^^c) _ a(hT Vd „ b±^~ (b±\^c)x(bTV~c)~ (py-ic^^ " ^* ~ *^ ^"^' 276 ELEMENTS OF ALGEBRA. II. If the Denominator is a Binomial Involving only Quadratic Surds. Multiply both terms of the fraction by the terras of the denominator with a different sign between them. Note. It is often useful to change a fraction which has a surd in its de- nominator to an equivalent one with a surd in its numerator. Thus, 8 SXVI 8 I'S^ J X 2.236+= 1,3416+. V5 V5 X V5 5 Exercise 99. Eationalize the denominators of: 2 3 2 - ^2 3 V5 1. V2 + V3' 2 V5 - V6 1 + V2 V3 + V2 8-5 V2 , 2 V"5 - V2 1 6 3- 2 a/2' V5 + 3V'2' 3-2 Vg' 'V^64 ' Vx — Vy , Sx — Vx y _ Va + a; + V<?' — x o. "s/x + Vy Va: y — 2>y Vet -\- x — Va X . X — Vx^ — 1 a 1 2 a 4. + Vx^-l' \/a+Vb V5-V'2' 3a/2^-^ Given V2 = 1.414, V3 = 1.732, V5 = 2.236 ; find the approximate values of: 5. ^_; V50; 8K288 '' ' ' 6. V2 ^ V5 2V675 V500 1 + V2 1-V5 3 1.1 2 + ^/2' 3+V5' 21/2-3^/3 ^5-^2 2 + ^3 RADICAL EXPUESSIONS. 277 113. Division of Snrds. 2 V54 -^ 3 ^6 = I VV = f X 3 =2. Ingeneral, a^i^6^y = ^^)" =^Vy* I. If the Divisor is a Monomial Reduce the surds to the same order (if necessary). Prefix the quotient of the rational factors to the quotient of the surd factors. ,- . ,- ,-x 3\/3 3 V3X (3^3-2^2) 3 V3 -r (3\/3 + 2 V2) = :^ p = 7 7= V\ / y A ^ ^ ^ 3'v/34-2V2 (3V3+2-v/2)x(3V3-2V2) ^27-6V6 Hence, in general, II. If the Divisor is a Binomial Involving only Quadratic Snrds. Express the quotient in the form of a fraction, and ration- alize its denominator. ^a -^ \/b - \/c) a + 2 \/ab + b-c {\^a+ \/b-\-^c. Divisor multiplied by '\/a, a + \/a b — \/a c First remainder, \/a 6 + 6 + V** c ~ c Divisor multiplied by y'ft, \/a b + b — ^/bc Second remainder, y'a c + ^b c — c Divisor multiplied by \/cj ^ac + \/6 c — c Hence, in genenil, III. To Divide a Polynomial by a Polynomial. Proceed as in Art. 33. 278 ELEMENTS OF ALGEBRA. Exercise 100. Simplify : 1. 21V384^8V98; 5 \/27 ^ 3 V24; \/l2^V^24. 2. - 13 VT25 -^ 5 V65 ; 6 Vl4 H- 2 ^21. 2V98 ' 7 a/22' 5V112 ' V394 ' ^2 * Vs' 4 IJ ^2| - I Vll; -^12 -- ^2 ; V6 -- A^4. 5. 20 ^^200 -4- 4 a/2 ; ^18 ^ a/6 ; 4 ^32 ^ '^IG. 3 a/108 5 a/14 15 a/84 7. (15 a/105 - 36 v'lOO + 30 A^81) -^ 3 Vl5, 8. '^OOei^ViO; a^'a^c-^^^; Va -^ \^. m — 71 ^ m — n ^ {m ~ nf '3 »2 10. {acx^Vy—hcy ^/x) -^ c a/^ ; '\/a~x -^ ^'o^. 11. ^4 m 7^2 -H V2w3^; v^2W^ X ^^^?^3^ aA;?^5 12. A^4^i2^XA^'9^^i2^*^v'25^^2^; <^d~^-r-\/^' 13. -y— ^x-V/-2--^— -V/-|— • (aj-l)-f-(A/aj-l). RADICAL EXPRESSIONS. 279 14. V10.4976 -^ 2 Vo ; (2 a: - Vo; y) ^ (2 Va: y - y). 15. (:^ a/3 + 2 V2) ^ (a/3 + V2) ; 4 ^a^ -f- 3 Vo^. ,^ 2A/T5 + 8 . 8V3+ 6a/5 8-4\/5 . 3a/5-7 Id. =:r- -7- — —', 7^7- -T- pr- • 5- a/15 5a/3-3a/5 1 + a/5 5 + a/7 17. (^x« + ^^v + w") - (V'^'^" - va;V + vy). 114. Involution and Evolution of Surds. l^v/ir=[Mi)T=i-.x®'=4v/I=^vi.- y486av/4a« = [3« X 20(220*)*]' = [3^ X 2^a^f = 3 X 2*0* = 3^/2^ m mp In general, (a^i -v^t^)^ = (a«i 6"*)' = a"!^ 6^ = a^h^ >v/6*^. Vo'»i v^6™ = (a"*! 6" j^ = a »• 6" ^ Hence, Express the surd factors with fractional exponents, and proceed as in Art. 104. Example 1. V A«/;5 2a) ~ Ul 2aJ -©■-3Gr(i).3@C4)'-e)' -^ -'S)©"K)(^)- ^? 2»a« 3a* 2cH 3q« 3 4a*c* 3 i«ra 8a» 8a«" 280 ELEMENTS OF ALGEBRA. 'i CO + I + + -0 Li i<5 ^ CO 1 1 1 y 1 '> 1 ^ + + -"« Ol CM II Th + Hw -* « c Tt^ ^ I HM -O CO 1 ^ % GO GO + + Hn c c -^ Tf '-Ca ^5 a r^ 'd S S c ^ 1 fn'' fi" .2 -U jT p-T a; J ^h" 'IS ir^ -M c s s ^ (V S id 1 S 1 i s s -tJ C 8 -^ ^ -TIJ -n -s 02 CK 03 ^ rt a G rt ^ ^ ti ^ £ •>-( o; S £ s pR pR '/2 m 02 02 RADICAL EXPRESSIONS. 281 Exercise 101. Find the values of the following : 1. mf; ^VE; i^Vlf; ^^2; (^32)'. 2. 'fe^; ^m-, i'^I^jf; V-^Gi. 3. {</uf; vQ^; mf; ^vff; mt 4 ^JV^; (^27)^ --(^A; (2^3¥f ; 'sp^- 5. (2 ^^6)'; y'V©"' '^"•^^■^"^ (2a^2Fo/. 6. ~^/•i-•a-'■, {Z'^Wc^f; ^iU^"; "v''27»=^. 7. 'v'»-»v'^»; [\/(a-c/]"; "-^1; [jv^I^^-^js, 11. v^9-a:— ; [{x ^ y) V^y^ - VnS^2i^^, Find the values of the following, and express the results in terms of positive exponents, by inspection : 12. (^^^T?f ; Wl + V\f; {V2 + Vsf. 282 ELEMENTS OF ALGEBRA. 13. (^3^l)MV^+^i)- ["^-fl- 15. M^.o]^ [(-^«--f +(fr^)-T- 16. [jn^ J'Zm 2-^m T ["^ /m-^ 4 -^^3a "[ 12 mV n "^ 4/n J ' L V ^4 ^^^^^3 J Find the square roots of : 17. ^ + 1 + 9 ^^ + 1^-1 _ 1^-1 ^^ _ 6 ^^. 18. 1 - 2"+^ + 4"; 9" — 2"+^ X 3" + 4". 19. r V2. ic 4?/ -^a; '2/ m 20. v"^ - 4''^2;5- + 4 :i:"^ + 2 a/^^ "' - 4 '-v^^^ '" + a/^' Miscellaneous Exercise 102. Find the values of the following : 1. 'V^; v^; V3xV27; 1""; "'^4; a/X^V|. 2.' ^^32^; V^e^m^^ V^m ^ V^ Reduce the following to their simplest forms : 18|' -^'2'- 3. t J^,; V^; ^3888; ^!^±^sj-^ ») RADICAL EXPRESSIONS. 283 Reduce to equivalent surds of the same lowest order : 5. V2, '^, •^5; -v^y, a!^8, 4/5; 3^75, 3v^54. 6. 2A/I8, ?,\/U, 4A!^r62, 5-^128; V^\ </V, i^^. Change to similar surds : 7. •^27, -^144; -C^Si, 3^; W2, ■^243. 8. ^02, VI; VTo, 4'Wb; I'^lh, V^. 9. 2\/^. ^7-26^, ly^^; K.1G; 4^275. 10. n, VJl, ^i^, 3 ^^; V20, 3 '^. 4 Vl25. 11. ^32. ^128; VM^. y/^'. SfW^'- 12. ^J^. ^192, ^S". ^^. ^A; ^2, ^2?. Arrange in order of magnitude : 13. 3 a/2^, 4 </^, 3 -^3 ; 5 ^8, 3 ^9, 3 VlO. 14 fnfV3~, iV5, 2^; Vi ^H- 15. a' -e^^sTjy, -^ ^(25^^-, (64)" sj"^, . Find the values of: 16. V243 4- \/27 4- v'48; 2 V^189 + S'l^'STS -7^, 284 ELEMENTS OF ALGEBRA. 17. 4 V5 X 'V^lT; 3 'v'^gOO -^ \/5; \J2 V 2Vl -^ ^V^^' 18. 5V'2 + 3V8-2V32; 3 -^^ST - 4 ^1^192 + ^648. 19. 'V^m X #432 ; I V5 X |- #2 x #80 x #5. 20. ^64+5-^32-^^108; i(^27 + | -V^i92 + •^81). 21. 2'v^JJ^ + a/60 - \/225 - Vf ; J V|^ { a/2 + 3 V| )• 22. (a^|^2^^)--V^I6; (-^9-2^21 + 4^1)2-^9. 23. (6 -^1 + ^18) -f- -^72 ; 1| -^/20 - 3 -v^5 - -v/f 24. J,a ^60-^(2 ^240 + 7^31); y/j^-^^^)*. 25. (-^ro -2-^4 + 4 -^54) (o -^64 + sV^- 2 ^32). Eationalize tffe denominators of: ^g V2Q - a/8 . (3+ V3)(3 4- #5) (#5 - 2), * V5 + V2 ' (5 - V5) (1 + VS) ^ m -\- (171 — Vvi^ + a? 71^ x"' 2 m — a ?l + V??i2 ^- ^2 7^2 „ ' ^5 _|_ /y/3 _ ^2 Simplify the following : 28. ^_ ; V(f|y" X V(||f ; #(8'a3&)2 x #(2 a &3)2. V2 ^^ 7 + 3 \/5 _^_ 7-3a/5 ^ /^9?»Ti x V3l<T" ^" 7-3^5 7 + 3A/5' V 3V3^ 30 . {a -r #«)" + ' y^(^ j/^"")"'; #^2^^""* X \x^yl RADICAL EXPRESSIONS. 285 31. 4^aX'i^:r^X ^a* X "/a X ^aV x ^^. 33. 4.x^<77xf x-^^^„-^(^y-^^/*^ 34. V(a>) ^ v/(-^.,p; 1^,; |(|)t. 35. .^- (g) ^ Vl^. ; I ^1 + 1 ^^' - 2 (S)«. 37 V^ + «^ V^ ^^ ^ 4 ^^ 38. ^-Hl^^^i -H f 1 + -^V; i^a^ + ^a)3. ^5 X ^^3 40. Express r^ with a single radical sign. Queries. What sign is given to the Titli ])ower ? To the nth root? Why? How change the order of a suixl ? In T., Art. 112, why take m less than n ? How rationalize a sunl denominator ? What powers of n^ative nuniWrs are positive ? What n^ative ? 286 ELEMENTS OF ALGEBRA, Imaginary Expressions. 115. An Imaginary Expression is an indicated even root of a negative expression ; as, V— a ; a -\-h V— 1. V— 1 is an imaginary square root ; a V— 1 is an imaginary fourth root; etc. -V^T^^ :^ ^a2 X (- 1) = V«^ X V^ = « a/^^- /^-TJ - .^6 X (- 1) - ^b X V-^- Hence, Every imaginary square root can he expressed as the product of a rational or surd factor multiplied by \/— 1. The successive powers of ^y/— 1 are found as follows -. )ip=(-l)i = + V=T; )*/=(-!) =-l; )*r = (-i)' = (-i)(-i)* = -V~; )J]'=(-l)^ = + l; )i]«=(.-l)» = -l; )i]»=(-l)^ = + l; )J]» = (- 1)1 = (- i)^(- i)i = + ^—; and so )*]'=(- 1)5 =VPT)= -±-v/^orTl, oc?c? or eyen integer. Hence, The successive powers of \/~ 1 form the repeating series : + V~h -h -V^» +1- The methods for operating with imaginary expressions are the same as those for surds ; but before applying the methods it is better to remove the factor /y/— 1. All cases of multiplication can be made a direct application of Arts. 97, 114. w- 1? = = [(-1 w- -xY- = [(-1 w- ~xY-. = [(-1 w- -lY- = [(-1 w- -^Y- = [(-1 [V- -xY-- = [(-1 [V- 1]' = - [(-1 w- T]' = = [(-1 w- T/ = = L(-i on. In general, W~ T]" = =[(-1 accordin gas n is an RADICAL EXPRESSIONS. 287 niustratioiiB. ^/- 6aH^= y/S a^b* X (-1) = \/S aH^ X \/-l = 2ab \/Tb X V^. V-9«^ + V-49a«-V4a'«= 3a ^- 1 +7a/v/- 1 -2 a = 10a V^- 2a = 2 a (5 V=n - 1). 3 >v/^ X 4 -v^^ = (3 V^ X \/-^)("* V^ X V-^) = 3 ^3 X 4 y 2 X ^/^ X V^ = 12V3X^X [(-l)*]' = - 12 ye. 2 -v/^ X 5/v/^ X 3 V^ = 2 V3 X 5 V2 X 3 >v/6 X \/^ X ^^^ X V"^ = 30>v/3 X 2 X 6 X [(- l)*]" = - 180 ^~l. = f -v/3 X 1 = J V^- _ (i + y-H)' _ i+.2v^_+ (-0 Example. Multiply 1-2 V"^ by 3 + ^y/^. ProcesB. 1—2 ^—~i 3+ V-1^ 1 - 2 -y/- 1 multiplied by 3, 3-6 V" ^ 1-2 y'lH! multiplied by y'^, 2 + V^ Sum of the partial products, 6 — 5 /y/— I 288 ELEMENTS OP ALGEBRA. Notes : 1. Imaginary expressions represent impossible'operations ; yet it is a mistalie to suppose that they are unreal, or that they have no importance. 2, If the student employ the method of multiplying or dividing the expres- sions under the radicals (Arts. Ill, 113), for all cases in multiplication and divi- sion, he cannot readily determine the sign of the product or dividend. Thus, V^^ xV—a = V—aX—a= Va^ = ±a. 3. Is the above product both ±a or — a ? We are limited to the considera- tion of the product of two equal factors, and we know that the sign of each is negative ; also, that Va^ = it «. Hence, the sign of Va^ will necessarily be the same as that of each of these factors. Therefore, it will be the same as was its root. Thus, V- 3 X 1/- 3 = - 1/9 = - 3, Exercise 103. Simplify : 1. V^; '^-16; V- 12 a; V^^T^; V^ 2. V-49a2-&6. ^^7729; ^IT^; y'^^^". Find the values of : 3. (V— i)i^,(V^f ; iV-if; {-V—lf. 4. i-V^lf; (-V=^r; i-V=-lf; {-V^f 5. A/-25 - A/-49 + V-121 - a/-64 + V-1- a/-36. V-22 V-216 6. 2 V- 24 + —= - V- 18 ; ^...^ - V-3 A/-33 V-324 7. V- 36 a^ 4- V- 9 a^ - V- (1 - af a^ - V- a\ 8. V-{ct-hf+ V-(a2- 2ab + b^)+ V-1 6 a^ b^-V- 4 a^ Multiply : 9. V^ by V^; 3 V^ + V^^ by 4 V^^ RADICAL EXPRESSIONS. 289 10. 2 V^ by 4 V'^; 1 + V^ by 1 + V^. 11. V- 2 + 3 V^ by V^^ + 3 V^. 12. 3-2 V-4 by 5 + 3 V^^; 4 + V^ by 4- V^. 13. 1 + V^ by 1- V-1; 2 - V=^ by 1 - 2 V^~3. 14. 2 V^ - 6 V^ by V^ + V^. 15. Va — ^ by V^ — a ; a + V— a; by a — V— a;. 16. a V— a + b V— b by a V— a — 5 V— 6. Divide : 17. V^^ by V^^; - \/^ by - 6 V^. 18. V^ by V- 20; V- 24 - V^ by a/^^ 19. 2 V— 4 «*-» by V— a^ \ a + V- a by V- a^. 20. - 2 V^ by 1 - \/^ ; 2 by 1 + V^. 21. \^-^^^ by v^- 5; '^^^^ by ^=^. 22. 4 + V^ l)y 2 - V^; V^3 by 1 - V^. Rationalize the denominators of : 23 ^i^J^^^- 2 \/:ri _ 3 yry 3 + 3 V^ " ' 2 - V^' 4 a/^ + 5 V^^' 2-2 V=I ' Queries. To what form can all imaginary monomials be reduced ? In multiplication and divi.sion why separate the imaginary expres- sions into their sunl and imaginary factors ? Is it necessary in all ? 19 290 ELEMENTS OF ALGEBRA. Quadratic Surds. 116. I. A quadratic surd cannot equal the sum or differ- ence of a rational expression and a quadratic surd. Proof. If possible, let ^/a = 6 db ^\fc, in which ^a and y^c axe dissimilar quadratic surds, and 6 a rational expression. Square both members, a = 6"^ ± 2 & ^/c + c. ± a T ^^ T c Transpose, ± 2 6 \/c = a — ft^ — c*. .♦. ^/c 26 That is, a surd equal to a rational expression, which is impossible. Therefore, ^\/a cannot equal h ± ^ c. II. i/" a + Vb = X + Vy, in which a and x are rational and Vb and Vy cire quadratic surds, prove that a = x and b = y. Proof. Transposing, /y/6 = (x — a) + V^?/- Now if a and a; were unequal, we would have a quadratic surd equal to the sum of a ra- tional expression and a quadratic surd, which, by L, is impossible. Hence, a = x. Therefore, ^Jh = ^^y, ot b = y. III. 7/" V a + Vb = Vx + Vy, prove that y a — Vb = Vx — vV- Proof. Square both members, a + \/b = x + 2 aJx y + y. Therefore IL, a = x + y {!) and a/6 = 2 ^/x^ (2) Subtract (2) from (1), a - \/b = x ~ 2 \/xy + y. Extract the square root, V a — \/b = y\/x — \/y. Similarly it may be shown that if V « — ^/b = ^/x — /y/jr, then V a + ^Jb = ^Jx + ^/y. RADICAL EXPRESSIONS. 291 Square Root of a Quadratic Surd. 117. To find the square root of a binomial surd a ± Vh. Process. Let Va ± a/6 = V^ ± Vi^ (1) Then (III., Art. 116), Va T V^ = V-^ T \/y (2) Multiply (1) and (2) together, ^af^ b = x- y (3) Square (1), a ± aA = x ±2 ^x y + y. Therefore (II., Art. 116), a=x-hy (4) Add (3) and (4), a + v^^"^ = 2 x. .-. x = ^ , a - Va* — b Subtract (3) from (4), a - V a* -b = 2y. .'. y = ^ Therefore, V7±Vb = \J ^^ V^^ ± \/" ~ ^"'~^ (0 HotM; 1. Evidently, unless d^ — h be a perfect square, the values of Vx and Vy will be com plex surds ; and the expression Vx + \ y will not be as simple as V a + 1/6. 2. Since, Va'^c + Vbc — \/c{a ^- Vh\ also if a^ - 6 be a perfect square the squats root of a + Vh may be expressed in the form Vx 4- Vy, the square root of Vcflc ■{■ Vbc'is of the form \'c ( V'x -f Vy ). 3. Frequently the square root of a binomial surd may be found by in- si>ection. Thus, FiTid ttoo numbers whose sum is the rational term^ and whose product is the square of half the radical term. Connect the square roots of these numbers by the sign of the radical term. Examples : 1. Find the square root of 3^ - VlO. Process. Let Vi - Vy = "^H - aAo (1) Then (III., Art. 116), y'i -f Vy = V^T'^ (2) Multiply (1) and (2) together, x-y= \/*^ _ lo = | (3) Square (1), x - 2 V^ 4- y = 3i - - /y/lO^ Therefore (II., Art. 116), z + y = 3| (4) From (.3) and (4), x = 2|, y=l. Therefore, ^3^ - \/Tb = a/| -I = ^ V^- 1- 292 ELEMENTS OF ALGEBRA. We may employ the general form (i). Thus (since a = 83 and y6 = + 12 V35), L^ ^ ^g^2 _ (|2 ^35)2 2. V83 + 12 ^/3b = \ ^^ — - 4/ 83 - V8'3^ - (12 V35f _ / 83 + 43 / 83 - 43 = ^63 + ^20 = 3 v^ + 2 Vs. 3. Va/ST - 2 a/6 = V V3 (3 - 2 a/2) rr: ^3 X a/3 - 2 a/2, ^ also \/3 - 2 a/2 (in which a = 3 and a/& = - 2 a/2) ^ i / 3 + V3-^ - (2 A/2y _ i / 3 - V3"^ (2 a/2)^ ^ ^ _ i .-. a/ a/27 - 2 Ve = a!^3 (a/2 - l). 4. Find by inspection the square root of 103 — 12 a/h. Solution. The two numbers whose sum is 103 and whose product is (^ — |— j , are 99 and 4. Hence, Vl03 - 12 \/Tl =: a/99 - a/4 = 3 a/iT - 2. 5. Similarly, VlO + 2 a/21 := a/7 + Vs, because 7 an^ 3 are the only numbers whose sum is 10 and whose product is (a/21) • ^ ^ Exercise 104. Find the square roots of: 1. 7-2A/rO; 5 + 2V6; 41-241/2; 2J + V^. 2. 18-8 V5 ; 11 + 2 VSO ; 13 - 2 a/42. 3. 15~V56; 47 -4 a/33; 6-2^5; 10 + 4 a/6. RADICAL EXPRESSIONS. 293 5. V27 + a/15 ; 2?7i + 1 + 2 Vm^ + 7i - 2. 6. (m^ + m) ?i - 2 vi n Vm ; 9 — 2 VU. 7. (wi + w)'-^ — 4 (m - /O A/m?i ; 3 a; - 2 a: V2. Find the fourth roots of: 8. 97-56V3; | a/5 + 3J ; 56 + 24 V5. 9. 17 + 12V2; 4(31 -8 Via); 248 + 32 V^. Simple Equations Containing Surds. 118. Examples : 1. Solve V4 ar^ - 7 x + 1 = 2 a; - 4 (1) Process. Square (1), 4x^-7x+l = 4x^-7\x + ^. .'. X = ]l\. Hence, To Solve an Equation containing a Single Surd. Arrange the terina bo as to have the surd alone in one member, and then raise each member to the power indicated by the root index. Note. If the equation contains two or more surds, two or more operations may be necessary in order to clear it of radicals. Thus, 2. Solve \/'2b x-1%- ^4x- 11=3 -^/i. Process. Transpose, \/25a:-29 = 3 ^/x 4- *J\x-\\ (1) Square (1), 25x-29 = 9x+6 V(4a:- ll)a:+4a:-ll. Transpose, etc., 'v/(4a;- 11) x = 2 x — 3 (ii) Square(2), 4x«- llz = 4a;«-12ar + 9. .-. x = 9. 294 ELEMENTS OF ALGEBRA. 3. Solve ^^-7= = -^ (I) \ X + n ^x + 3n ^ ' Process. Clear (1) of fractions, transpose and unite, etc., / ^ r TT /- mn ( mn \2 (m — n) \/x = mn. Hence, \/x = . .*. x = ^ . \/m + X 4- ym — x 4. feolve ^' , J — n. \m -\- X — 's/m — X Process. Rationalize the denominator, fn. + 's/m'^ — x'' (1) From (1), y'rn^ - x"^ - nx - m (2) Square (2), m^ ~ x^ = n^ x^ -"Imnx ■\- m\ Transpose, etc., a;^ (l + n^) =r 2 mna:. Divide bv a;, a:(l + w^) =: 2mn. .*. a: i= 7— — ;, Exercise 106. Solve : 1. V^ + 5 = 4 ; V3 ^^ + 6 = 6 ; ^x^-2^x^% 2. Va:2 _ 3 ^. _}_ 5 ^ :i; _ 1 ; ^2^-3 -1=2. 3. V'3 + V4 + V^^^= 2; Vr+W^^m = a: + 2. 4. ^/mx^~a = 'v/c^TT ; V^rr2 + ^4 - ^3^ = 2:. 5. VJT~2 =. 2 V'2ic-3 ; V3 a: + 5 = 3 ^"Ix ^ 1. 6. V3.r- 4= V2a: + 16; ^^2 ^' - 4 = V 4 - V2 rr. 7. 3Vi = RADICAL EXPRESSIONS. 8 295 V9 a; - 32 4- V 9 a; - 32. 8. Va: + 3 4- Vx + S - V4 :c + 21 = 0. 9. ^Vx+ - ^Vx — 5 = y/2 Vi. Q 10. 'V^m^ -f a; Vn^ + x^ = Vx-{- vi; Vx + Vx—2 = —r' \ X 11. A^4+2V2^-5 = V3; V| + :^^m_ V a: — V m ^ V2 ic + 1 + 3 Va; _ Vm a; — n __ 3 v m a; — 2 7i a/2 a; + 1 — 3 Va; Vwi x-\- n 3 Vw x + 5n 13 V5a;+ VB Va;4- 5 V6a;+2 _ 4:V6x + 6 V3^ 4- V3 ~ V^ + 3' V6^- 2 ~ 4 V6'^ - 9 14. v/^+v/^^=c/4^- "w + a; 'm — a; »7?r — ar Solve the following for x and y : 15. a; + 4 V3 + y = 15 — a; 4- y V5. 16. a; 4- y 4- a; Va 4- y V^ = 1 — Va. 17. a: - 5 + (2 y - 3) V3 = 5 a; - Vl2. 18. a; — rt 4- {y — 3) Va 4- 2> = w a; 4- Vol. 19. a — Va; + y = y — x — Vm 4- 71. 20. a; Vw(Vm4- 1) = w — wi4- y V^(l — V^. 296 ELEMENTS OF ALGEBRA. CHAPTEK XX. LOGARITHMS. 119. If a^ = m, then / is called the logarithm of m to the base a. Hence, A Logarithm is the exponent by which a certain num- ber, called the base, must be affected in order to produce a given number. The logarithm of m to the base a is written logam. Thus, log„m = I expresses the relation a^ = m; logj^ 100 = 2 expresses the relation 10^ = 100, etc. Since numbers are formed by combinations of tens, any number may be expressed, exactly or approximately, as a power of 10. Thus, 1000 = 103 . ei-c 120. Common System of Logarithms. This system has 10 for its base, and is the only one used for practical calculations. Thus, Since 100=1, log 1 = 0; since 10^ = 10, log 10 = 1 ; since 102 = iqO, log 100 = 2 ; since lO^ = 1000, log 1000 =: 3; since 10* = 10000, log 10000 = 4 ; and so on. Since lO-i = J^ = .1, log .1 = - 1 = 9 - 10 ; since 10- 2 = ^i^ = .01, log .01 = - 2 = 8 - 10 ; since lO-s = ^Jq^ = .001, log .001 = - 3 = 7 - 10 ; and so on. It is evident that the logarithm of all numbers greater than 1 is positive J and of all numbers between and 1 is negative ; also, that the logarithm of any numbers between LOGARITHMS. 297 1 and W is -f a fraction; 10 and 100 is 1 + a fraction ; 100 aiid 1000 is 2 + a fraction ; 1 and .1 is — 1 -f a fraction, or 9 + a fraction — 10; .1 and .01 is — 2 + a fraction, or 8 + a fraction — 10; .01 and .001 is — 3 + a fraction, or 7 + a fraction — 10; and so on. It thus appears that the logarithm of a number consists of an integral part, called the characteristic, and a I'ractional part, called the mantissa. The mantissa is always made positive. IUu8tration8. It is known that log 5 = 0.69897; log 12 = 1.07918; log 2912 = 3.4(3419; etc. These results mean that loo«»8»' = 5; 1O1.07918 ^ 12; 108.4M19 = £912 ; CtC. Notes : 1 . 'Hie fractional part of a logarithm cannot be expressed exactly, but an apiiroxiniate value may be found, true to as many decimal places as desire^l. Thus, the logarithm of 3 is found to be 0.477121, true to the sixth place. 2. For brevity the expression "logarithm of 3" is written log 3. The expression "log «" is read "logarithm of x." 3. Logarithms were inventetl by John Napier, Baron of Merchiston, Scot- land, and first published in 1614. 4. Ih-re are only two systems of logarithms in general use : the Natural, or Hyperbolic, system, and the lirhjtjsian, or Common, system. The base sub- script of the former is e, and that of the latter is 10. 5. The natunil system, invente<l by Jolm Speidell and published in 1619, is employetl in the higher branches of analysis and in scientific investigations ; its base is 2.718281828+ . 6. The common system, more properly called the denary or decimal sys- tem, was invente<i by Henry Brigps, an Engli.sh geometrician, and first pub- lished in 1617. The logarithm of its base, 10, is alrmys 1. 7. The logarithms invente<l by Napier are entirely different from those in- ventetl by Si>eidell, though they are closely connected with them. The natural system may be regardeil as a modification of the original Napierian system. 121. Since log 1 = 0, log 10 = 1, log 100 = 2, log 1000 = 3, etc., the characteristic of the logarithms of all numbers consisting of one 298 ELEMENTS OF ALGEBRA. integral digit (that is, all numbers with one figure to the left of its decimal point) is 0; of all numbers consisting of two integral digits is 1 ; of all numbers consisting of three integral digits is 2 ; and so on. Hence, I. The characteristic of the logarithm of an integral number, or of a mixed decimal, is one less than the numher of integral places. Since log .1 = - 1, log .01 = - 2, log .001 = - 3, etc. ; the charac- teristic of the logarithm of any decimal whose first significant figure occupies the first decimal place (that is, of any number between 0.1 and 1) is — 1 ; of any decimal whose first significant figure occupies the second decimal place (that is, of any number between 0.01 and 0.1) is — 2 ; of any decimal Vfho^Q first significant figure occupies the third decimal place (that is, of any number between 0.001 and 0.01) is — 3; and so on. Hence, II. The characteristic of the loga7nthm of a decimal is negative, and is numerically equal to the numher of the place occupied hy the first aignificant figure of the decimal. The characteristic only is negative. Hence, in the case of decimals whose logarithms are negative, the logarithm is made to consist of a negative characteristic and a positive mantissa. To indicate this, the minus sign is written over the characteristic, or else 10 is added to the characteristic and the subtraction of 10 from the logarithm is indicated. Thus, log .0012 = 3.0792, or 7.0792 - 10 ; read "characteristic minus three, mantissa nought seven ninety-two," or "characteristic seven minus ten, etc." In reading the mantissa, for brevity, two inte»- gers are read at a time. Thus, log 2 = 0.30103, is read "the loga- rithm of two equals characteristic zero, mantissa thirty ten three." Illustrations. The characteristic of the logarithm of 9 is 0; of 32 is 1 ; of 433 is 2 ; of 39562 is 4; of 632.526 is 2 ; of .42 is - 1 ; of .023622 is - 2 ; of .0000325 is - 5 ; etc. LOGARITHMS. 299 122. Let m and n be any two numbers whose logarithms are x and y in the common system. Then l(F = in and 10*= n. Multiply- ing the equations together, we have 1(F+*' = mn. Hence (Art. 1 19), log mn = z + y. But x = log m and y = log n. Therefore, log m n = log m -H log n. Similarly log in np = log m -f log n + log p ; etc. Hence, Tlie logarithm of a product is found by adding together Die logarithms of its factors. Ulustrationa. Given l(»g 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, log 7 = 0.8451. log 252 = log (2 X2X3X3X7) = log 2 + log 2 + log 3 -f log 3 + log 7 = 2 log 2 + 2 log 3 + log 7 = 2 X 0.3010 + 2 X 0.4771 + 0.8451 = 0.6020 + 0.9542 + 0.8451 = 2.4013. log 300 = log (2 X 3 X 5 X 10) = log 2 + log 3 + log 5 + log 10 = 0.3010 + 0.4771 + 0.6990 + 1 t= 2.4771. Exercise 106. Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, log 7 = 0.8451 ; find the values of the following : 1. log 6; log 64; log 14; log 8; log 12; log 15; log 84. 2. log 343; log 16; log 216; log 27; log 45; log 36. 3. log 90; log 210; log 3600; log 1120; log 1680. 123. If any number be multiplied or divided by any integral power of 10, since the sequence of the digits in the resulting number remains the same^ the mantis-sa) of their logarithms will be unaffected. Thus, since it is known that log 577.932 = 2.7619, 300 ELEMENTS OF ALGEBRA. log 5779.32 = log (577.932 X 10) = log 577.932 + log 10 2.7619 + 1 = 3.7619. log 57793.2 = log (577.932 X 100) = log 577.932 + log 100 2.7619 + 2 = 4.7619. log 57.7932 = log (577.932 X 0.1) = log 577.932 + log 0.1 2.7619 + (- 1) rr 1.7619. log 5.77932 = log (577.932 X 0.01) = log 577.932 + log 0.01 2.7619+ (-2) = 0.7619. log .577932 = log (577.932 X 0.001) = log 577.932 + log 0.001 = 2.7619 + (-3) =1.7619. Etc. Hence, The mantissce of the logarithms of numbers having the same sequence of digits are the same. Illustrations. If log 44.068 = 1.6441, log 4.4068 = 0.6441, log .44068 = 1.6441 or 9.6441 - 10, log .000044068 = 5.6441 or 5.6441 - 10, log 440.68 = 2.6441, log 440^800 = 6.6441, etc. If log 2 = 0.3010, log .2 = 1.3010, log .02 ^ 2.3010, log 20 = 1.3010, etc. Hence, The mantissa depends only on the sequence of digits, and the characteristic on the position of the decimal point. Exercise 107. 1. Write the characteristics of the logarithms of : 12753; 13.2; 532; .053; .2; .37; .00578; .000000735; 1.23041. 2. The mantissa of log 6732 is .8281, write the logarithms of: 6.732; 673.2; 67.32; .6732; .006732; .000006732. . 3. Name the number of digits in the integral part of the numbers whose logarithms are: 5.3010; 0.6990; 3.4771. LOGARITHMS. 301 4. Name the place occupied by the first significant fig- ure iu the numbers whose logarithms are: 4.8451; 0.7782. Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.G990, log 7 = 0.8451; find the logarithms of the following number : 5. .18; 22.5; 1.05; 3.75; 10.5; 6.3; .0125; 420. 6. .0056; .128; 14.4; 1.25; 12.5; .05; .0000315. 7. .3024; 5.4; .006; .0021; 3.5; .00035; 4.48. 124. Let 7« be any number whose logarithm is x. Then UF^wi. Raising both members to the pih power, we have 10'" = inP. Hence (Art. 119), log mf = px. But x — log?n. Therefore, log Tn** = -p log m. Similarly log wi^'n' = p log m -\- q log n, etc. Hence, Tlie logarithm of any power of a number is found h/ multiplying the logarithm of tlie number by the exponent of Vie power. niustrations. log 5" = 10 log 5 =J0 X 0.6990 = 6.99(H), log .003* = 5 log .(K)3 = 5 X 3.4771 = 13.3855. log 864 = log 26 X 3« = 5 log 2 + 3 log 3 = 5 X 0.3010 + 3X0 4771 = 1.5060. Note. If the number i.s a decimal and the exponent positive, the j)roduct of the characteristic and exponent will be negative, and since the mantissa is made positive, we must algebraically add whatever is carried from the niantis.sa. Thii.«», log .0005" = 10 X 4.6990 = 40 + G.9900 = 34 + 0.9900 = 34.9900. Exercise 108. Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, log 7 = 0.8451 ; find the logarithms of: 1. 2*; 53; 7^; 8^ 3^; 64; 81; 72; (8.1)7; (2.10)6 2. 343; .036; .000128; (.0336)1^ (.00174)2; (3.84)» 302 ELEMENTS OF ALGEBRA. 125. Let m and n be any two numbers whose logarithms are x andy. Then 10^ = m and 10^' = w. .-. 10^-2' = m -f n. m " mn log - = a: - 2/ = log m - log n. Similarly, log ^^^ = log m + log n — (log wij + log Uj). Etc. Hence, The logarithm of a quotient is found by subtracting the logarithm of the divisor from the logarithm of the dividend. Illustrations, log | = log 3 -log 2 = 0.4771-0.3010 = 0.1761. log f = log 5 - log 7 = (0.6990) - 0.8451 = (1.6990 - 1) - 0.8451 = 0.8539 - 1 = 1.8539. Note. To subtract a greater logarithm from a less logarithm. Add to the characteristic of the minuend the least number which will make the minuend greater than the subtrahend ; also indicate the subtraction of the same number from the minuend so increased. Then proceed as before. Thus, log =^ = log 252 - log 300 = (2.4014) - 2. 4771 = (3.4014-1) - 2. 4771 = 1.9243. log '-^ = (3.6990) - 2T8451 = (T.6990 - 2) - 2.8451 = 0.8539-2 = 2.8539 or 8.8539 - 10. Exercise 109. Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, log 7 = 0.8451 ; find the logaritlims of: 1 ^ 2' _3^ 7 3 .003 .005 /SV .007 ' .05' 5' 5' 2' Q7 ; 02 ' VlO/ ' -02' 2 42. -:!!_. 125- '^- ^- ^i^. 5. ^ ■ ^' .0052' ' 5 ' 8.1' .000027' ' .007* 126. Let m be any number of which the logarithm is x. Then X 1(F = m. Taking the rth root of each member, we have 10'" = \/m. .'. (Art. 119), log \/m — - = -^ — . Similarly, log ^^m n LOGARITHMS. 303 The logarithm of any root of a numher is found hy divid- ing the logarithm of the number by tJie index of the root. «/- log 5 0.6990 ^ _„ IHuatrations. log ^ = -|- = — 5— = 0.1398. ,, log.0(X)7 4.8451 3.8451 + 7 ^„ - -,,^« log '^/ioOO? = -^ = -y— = 7-^ = 0.5493+1 = 1.5493. 1 VT-^. logl5<^ 5 log 3 X .5 5 (log 3 + log .5) logVl-5»=— ^r— = e = 6 ^5(0.4771 + 1.6990)^^^^^^^ o Note. If a negative characteristic is not exactly divisilile by tlie index of the root, subtract from the characteristic the least positive number which will make it so divisible. Indicate the addition of the characteristic so formed to the mantissa, and prefix the number subtracted from the characteristic to the mantissa. Then divide separately. Thus, log V75 = ?5|:5 = I:^ = '•6»y + ^ = 0.8495 + T = T.8495 or 9.8495 - 10. Ik ii it Exercise 110. Given log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, log 7 = 0.8451 ; find the logarithms of: 1. ^7; \^l; \/2- v^.^; ^243; ^12^; \^M; ^^. 2. ^^iTK2f; 5ix3i; ^,; g; ^^^; 6i x 3f. ,, ^^x\/2 j^^ ^ 7/ J.21)2_ V^2^ "• -;/l8xV2^ ^15' -^!^7' V(.()()084)2' ^^^^^fg ' 127. Table of Logarithms. The table (pages 304 and 305) gives the nianti.Hsae of the logarithms to four decimal i>]jices for all numbers from 1 to 10(X) inclusire. The characteristic and decimal points are omiUedy and must be supplied by inspection (Art. 121. 304 ELEMENTS OF ALGEBRA. N 1 2 3 4 6 6 7 8 9 10 11 12 13 14 ouoo 0414 0792 1139 1461 0043 0453 0828 1173 1492 0086 0492 0864 1206 1523 0128 0581 0899 1239 1558 0170 0569 0934 1271 1584 0212 0607 0969 1803 1614 0253 0645 1004 1335 1644 0294 0682 1038 1367 1673 0384 0719 1072 1899 1703 0874 0755 1106 1480 1732 15 16 17 18 19 1761 2011 2304 2553 2788 1790 2068 2330 2577 2810 1818 2095 2355 2(501 2833 1847 2122 2380 2625 2856 1875 2148 2405 2648 2878 1908 2175 2430 2672 2900 1931 2201 2455 2695 2928 1959 2227 2480 2718 2945 1987 2253 2504 2742 2967 2014 2279 2529 2765 2989 20 21 22 23 24 3010 8222 8424 3617 3802 ;;o32 3243 3444 3636 3820 3054 8263 8464 3655 3838 3075 3284 3488 3674 3856 3096 3304 8502 3692 3874 3118 3324 3522 8711 3892 3139 3845 8541 3729 3909 3160 3365 3560 3747 3927 3181 3385 3579 3766 3945 8201 3404 3598 3784 3962 25 26 27 28 29 3979 4150 4814 4472 4624 8997 4166 4330 4487 4639 4014 4183 4346 4502 4654 4031 4200 4362 4518 4669 4048 4216 4378 4533 4683 4065 4282 4393 4548 4698 4082 4249 4409 4564 4713 4857 4997 5132 5263 5891 4099 4265 4425 4579 4728 4116 4281 4440 4594 4742 4133 4298 4456 4609 4757 30 31 32 33 34 4771 4914 5051 5185 5315 4786 4928 5065 5198 5328 4800 4942 5079 5211 5340 4814 4955 5092 5224 5353 4829 4969 5105 5237 5866 4843 4983 5119 5250 5378 4871 5011 5145 5276 5403 4886 5024 5159 5289 5416 4900 5038 5172 5302 5428 35 36 37 38 39 5441 5568 5682 £798 5911 5453 5575 5694 5809 5922 5465 5587 5705 5821 5933 5478 5599 5717 5882 5944 5490 5611 5729 5848 5955 5502 5623 5740 5855 5966 5514 5635 5752 5866 5977 5527 5647 5763 5877 5988 5539 5658 5775 5888 5999 5551 5670 5786 5899 6010 40 41 42 43 44 6021 6128 6282 6335 6435 6031 0138 6243 6345 6444 6042 6149 6253 6355 6454 6053 6160 6263 6365 6464 6064 6170 6274 6375 6474 6075 6180 6284 6385 6484 6085 6191 6294 6395 6493 6096 6201 6304 6405 6508 6107 6212 6314 6415 6513 6117 6222 6825 6425 6522 45 46 47 48 49 6532 6628 6721 6812 6902 6542 6637 6730 6821 6911 6551 6646 6789 6830 6920 6561 6656 6749 6889 6928 6571 6665 6768 6848 6937 6580 6675 6767 6857 6946 6590 6684 6776 6866 6955 6599 6693 6785 6875 6964 6609 6702 6794 6884 6972 6618 6712 6803 6893 6981 50 51 52 53 64 6990 7076 7160 7243 7324 6998 7081 7168 7251 7382 7007 7093 7177 7259 7340 7016 7101 7185 7267 7348 7024 7110 7198 7275 7356 7033 7118 7202 7284 7364 7042 7126 7210 7292 7372 7050 7135 7218 7300 7380 7059 7148 7226 7308 7388 7067 7152 7285 7316 7396 LOGARITHMS. 305 N 1 2 3 4 6 6 7 8 9 55 56 67 68 69 7404 7482 7559 lOU 7709 7412 7490 7566 7642 7716 7419 7497 7574 7(W9 7723 7427 7505 7582 7657 7731 7435 7513 7689 7664 7738 7448 7520 7597 7672 7745 7451 7528 7004 7079 7752 7459 7636 7612 7686 7760 7466 7643 7610 7694 7767 7474 7661 7627 7701 7774 00 61 62 63 64 7782 7853 7924 7993 8062 7789 7800 7931 8000 8069 7796 7868 7938 8007 8075 7803 7875 7945 8014 8082 7810 7882 7952 8021 8089 7818 7889 7959 8028 8096 7825 7896 7966 8035 8102 7832 7903 7973 8041 8109 7889 7910 7980 8048 8116 7846 7917 7987 8056 8122 65 66 67 68 69 8129 8195 8261 8325 8388 8136 8202 8267 8331 8895 8142 8209 8274 8338 8401 8149 8215 8280 8344 8407 8156 8222 8287 8351 8414 8162 8228 8293 8357 8420 8169 8236 8299 8363 8426 8176 8241 8306 8370 8432 8182 8248 8812 8376 8430 8189 8264 8819 8382 8446 70 71 72 78 74 8451 8513 8573 8633 8692 8457 8519 8579 8639 8098 8463 8525 8586 8645 8704 8470 8531 8691 8651 8710 8768 8825 8882 8938 8993 8476 8637 8697 8657 8716 8482 8643 8603 8663 8722 8488 8549 8609 8669 8727 8494 8555 8015 8675 8733 8600 8661 8621 8681 8739 8606 8667 8627 8686 8746 75 76 77 78 79 8761 8808 8866 8921 8976 8756 8814 8871 8927 8982 8702 8820 8876 8932 8987 8774 8831 8887 8943 8998 8779 8837 8893 8949 9004 8786 8842 8899 8954 9009 8791 8848 8904 8960 9015 8797 8864 8910 8966 9020 8802 8869 8916 8971 0025 80 81 82 83 84 9031 9086 9138 9191 9243 9036 9090 9143 9196 9248 9042 9096 9149 9201 9263 9047 9101 9164 9206 9268 9053 9100 9150 9212 9263 9058 9112 9106 9217 9269 9063 9117 9170 9222 9274 9069 9122 9176 9227 9279 9074 9128 9180 9232 9284 9079 9183 0186 0238 9289 85 86 87 88 89 9294 9346 9396 9446 9494 9299 9360 9400 9460 9499 9304 9355 9405 9456 9604 9309 9360 9410 9460 9600 9316 9365 9416 9465 9613 9320 9370 9420 9469 9618 9325 9375 9426 9474 9628 9330 9380 94:^ 9479 9528 9836 0386 0435 9484 9533 9840 9390 9440 9489 9638 90 91 92 93 94 9642 9690 9688 9686 9731 9647 9696 9643 9689 9786 9652 96C0 9647 9694 9741 9667 9606 9662 9699 9746 9662 9609 9667 9703 9760 9666 9614 9661 9708 9764 9571 9019 9666 9713 9759 9576 9624 9671 9717 9763 9581 9628 0675 9722 9708 9686 9633 9680 9727 9773 95 96 97 98 99 9777 9823 9868 9912 9966 9782 9827 9872 9917 9961 9786 9832 9877 9921 9966 9791 98:^6 9881 9926 9969 9796 9841 0886 0930 9074 0800 9846 0890 9984 9078 9805 9850 9894 9083 0809 9864 0809 0943 0987 0814 9859 91K)3 9948 99*^1 9818 0863 9908 0962 0906 20 a06 ELEMENTS OF ALGEBRA. Explanation of Table. The left-hand column, headed N, is a column of numbers. The figures O, 1, 2, 3, 4, 5, 6, 7> 8, 9, opposite N at the top of the table, are the right-hand figures of num- bers whose left-hand figures are given in the column headed N. The figures in the column which they head are the corresponding man- tissse of the logarithms of the numbers. 128. To Find the Logaritlmi of a Number. I. Consisting of one Figure. The mantissae of the logarithms of single digits, 1, 2, 3, 4, etc., are seen opposite 10, 20, 30, 40, etc., and in the column headed O. To the mantissa prefix the character- istic and insert the decimal point. Thus, log 6 = 0.7782. log .6 = 1.7782. log 8 = 0.9031. Similarly, since the mantissa of log .009 is the same as the man- tissa of log 9, log .009 - 3.9542. II. Consisting of t-wo Figures. In the column headed N look for the figures. In the line with the figures, and in the column headed 0, is seen the mantissa. Then proceed as before. Thus, log 13 =1.1139. log 2.5 = 0.3979. log .92 = 7.9638. Similarly, log .00092 = 4.9638. III. Consisting of three Figures. In the column headed N, look for the first two figures, and at the top of the table for the third figure. In the line with the first two figures, and in the column headed by the third figure, is seen the mantissa. Then proceed as before. Thus, • log 313 = 2.4955. log 17.9 = 1.2529. log .279 = T.4456. Similarly, log .000718 = 4.8561. IV. Consisting of more than three Figures, Take the man- tissa of the logarithm of the first three figures as given in the table. Prefix a decimal point to the remaining figures of the number, and multiply the result by the tabular * difference. Add the product to * The Tabular difference is the difference between the two successive raan- tissjfi between which the required, or given, mantissa Ues. LOGARITHMS. 307 the mantissa thus taken. Prefix the characteristic and insert the decimal point as before. Thus, 1. Find the logarithm of 80.672. The tabular mantissa of the logarithm of 806 is 9063 The tabular mantissa of the logarithm of 807 is 9069 Therefore, the tabular difference = 6 The number 80672 being between 80600 and 80700, the mantissa of its logarithm must be between 9063 and 9069. An increase of 100 in 80600 causes an increase of 6 in the mantissa of the logarithm of 80600. Therefore, an increase of 72 in 80600 will produce an increase of ^ of 6 (or .72 X 6), or 4.32, in the mantissa of the logarithm of 80600. Hence, the tabular mantissa of log 80672 mtist be 9063 + 4, or 9067. Prefixing the characteristic and inserting the decimal point, we have log 80.672 = 1.9067. Similarly, since the mantissa of log .0005102 is the same as the mantissa of log 5102, 2 To find the logarithm of .0005102. The tabular mantissa of log 510 is 7076 The tabular mantissa of log 511 is 7084 .*. the tabular diflference = 8 Hence, the tabular mantissa of log 6102 must be 7076 -f .2 X 8, or 7078. .-. log .0005102 = 4.7078. Exercise 111. Find by means of the table the logarithms of the following : 1. 70; 102; 201; 999; .712; 3.6; .00789; 3.21. 2. .0031; .0983; .00003; 10.08; 29461; 3015.6. 3. 32678; V337; ^/Msm2; 4^| ; (.098x85)*. 308 ELEMENTS OF ALGEBRA. 129. To Find a Number when its Logarithm is Given. I. If the Given Mantissa is Found in the Table. The first two figures of the required number will be seen on the same line with the mantissa and in the column headed N, the third figure will be seen at the head of the column in which the mantissa is found. Finally insert the decimal point as the characteristic directs. Thus, 1. Find the number whose logarithm is 1.9232. Look for 9232 in tbe table. It is found on the line with 83 and in the column headed 8. Therefore, write 838 and insert the deci- mal point. Hence, the number required is .838. II. If the Given Mantissa cannot be Found in the Table. Find the next less mantissa, and the corresponding number ; also find the tabular diff'erence. Annex the quotient of the difference between the given mantissa and the next less mantissa divided by the tabular difference, to the corresponding number ; then proceed as before. Thus, 2. Find the number whose logarithm is 2.7439. The next less mantissa is 7435, corresponding to 554. The next greater mantissa is 7443, corresponding to 555. .-. the tabular difference = 8. The diflference between the given mantissa and the next less man- tissa is 4. Since the given mantissa lies between 7435 and 7443, the corresponding number must lie between 554 and 555. An increase of 8 in the mantissa causes an increase of 1 in 554. Therefore, an in- crease of 4 in the mantissa will produce an increase of ^, or .5, in 554. Hence, the mantissa 7439 must correspond to the number 554+ .5, or 554.5. Therefore (II, Art. 121), write 05545 and prefix the decimal point. Hence, the number required is .05545. 3. Find the number whose logarithm is 3.1658. The next less mantissa is 1644, corresponding to 146. The next greater mantissa is 1673, corresponding to 147. .•. the tabular difference = 29. The difference between the given mantissa and the next less man- tissa is 14. Annex \^, or .48 nearly, to the number 146, and insert the decimal point as the characteristic directs. Hence, the number required is 1464,8. LOGARITHMS. 309 Exercise 112. Find the numbers whose logarithms are : 1. 3.4683; 2.4609; 4.8055; 0.4984; 0.1959. 2. 3.6580; 2.4906; 4.5203; 2.5228; 0.6595. 3. 0.8800; 1.7038; 5.8017; 3.1144; 5.7319. 130. An Exponential Equation is one in which the expo- nent is the unknown number ; as, iif = n, ifrf = n. Such equations usually require logarithms for their solutions. Example 1. Solve the equation 2F = 1.5. Process. Take the logarithm of each member, x log 21 = log 1 .6. By means of the table, 1.3222 x = .1761. .1761 Therefore, x = . ^^aa = .1332, nearly. Example 2. Find the value of 3.208 X .0362 X .15734. Process, log (3.208 X .0362 X .15734) = log 3.208 + log .0362 + log .15734. log 3.208 =0.5062 log .0362 =2.5587 log .16734 = 1.1969 2.2618 = log .01827. Therefore, 3.208 X .0362 X .15734 = .01827. Example 3. Find the fifth root of .05678. Process, log .05678 = 2.7542. 5)2.7542 = 5)3.7542 + 5 .7508 + T = T.7508 = log .6634, nearly. Therefore, -y/.05678 = .5634, nearly. 310 ELEMENTS OF ALGEBRA. Example 4. Find the value of log^^ 144. Solution. To find loggi/g 144, is the same as solving (Art. 119) (21/3)' = 144, for I, squaring each side, etc., I = 4. Therefore, logg ^3 144 = 4. Exercise 113. Find by logarithms the values of the following : 1. 360 X. 0827; 117.57 X .0404 ; i^ ; (31.89)3 2. ^951; 380.57 X .000967; ^(•"^^^);^.y-Q"°^^^^ 212.6 X 30.2 7435 -^343 84.3 X 3.62 X .05632' 38731 X .3962' ^f2^' 4. — ■ • ^ ; 72132 X .038209 ; -7.000313. ^385.67 5. (61173)*; -^; ^; ^X^.00l; Ip. ^ ^ ' (.19268)i v'27 5^49 Solve the followiug equations : 6. 20" = 100 ; 2" = 769 ; 10" = 4.4 ; {^Y = 17.4. 7. 10^ = 2.45 ; 5'-'" = 2"+^ ; a/S^^^ = '^/W^. 8. 2* X 6*-2 = 52* X 7'-" ; 3^-' = 5 ; 4" = 64. 9. (1)" =10; rrf = n\ m"*+* = 71 ; ?/i""' X c^"' = n. 10. 2^+^ = 6^ 3^ = 3x2^+^ 31-0^-1^ = 4-1.^ 2^*-^ = 3^^-''. 11. a2*^3y = ^5 ^3x^2. :^ ^^0. ^x,^5y = (^7)4 ^^^ = (^y)8 LOGARITHMS. 311 Find the number of digits in the values of: 12. 312x28; 2^4; W^o . (4375)8. (396000)io. Find the number of ciphers between the decimal point and the first significant figure in the values of: 13. (.2)*; (.5y«>; (.05)5; (.0336)io ; "x/sm, 14. Given log x = 2.30103, find log xi Find the values of : 15. loga 4 ; loga 8 ; log^ 32 ; log^ 128 ; log, 1024. 16. log2 J ; log2 J ; logj ^ ; log^ ^^ ; log^ \^16. 17. logs 729; log5l25; log, 625 ; log, 15625; log, J. 18. log_e 1296 ; log_. - ^{^ ; log. ^{-^ ; logg^ 512. 19. logs ^5- 125; log848 49; log8l28; loga^/s tJi- 20. log,|/a^^^; log27^\; logj4; log, a ; log„|. 21. If 8 is tlie base, of what number is § the logarithm? Of what J ? Of what 1 ? Of wliat 2 ? Of what 3 ? Of what If ? Of what 2 J ? Of what 3 J ? Of what Y ^^ ? 22. In the systems whose bases are 10, 3, and J, of what numbers is — 5 the logarithm ? Find the bases of the systems in which : 23. log 81 = 4; log 81 = - 4 ; log j^^ = 4; log iiftW = -4; log i^ = ± 2; log 1024- = ±571. 312 ELEMENTS OF ALGEBRA. CHAPTER XXI. QUADRATIC EQUATIONS, 131. A Quadratic Equation is an equation in which the square is the highest power of the unknown number. A. Pure Quadratic Equation is an equation which contains only the square of the unknown number; as, 5 x^ = 17. An Affected Quadratic Equation is an equation which contains both the square and the first power of the un- known number ; as, 5 ^^ — 2 a^ = 10. o , ^"^ + 5 ^ a; 17 Example. Solve — I ^ = o + "t- • Process. Clearing of fractions, 12 a;^ + 60 — 9 a;^ = 4 a:^ -f 51. Transposing and uniting, x- = 9. Therefore, extracting the square root,* x = ± S. Hence, To Solve a Pure Quadratic Equation. Find the value of the square of the unknown number by the method for solving a simple equation, and then extract the square root of both members. Note. * In extracting the square root of both members of the equation a;2 z= 9, we ought to prefix the double sign (±) to the square root of each mem- ber; but there are no new results by it, and it is sufficient to w)1te the double sign before one member only. Thus, if we write ±, x = ± S, we have + x = 4-3, -^ X = — S, — aj = -j-3, and — x = — 3; but the last two become identical to the first two on changing the signs of both members. So that in either case, x = 3, and a? rr — 3. QUADRATIC EQUATIONS. 313 Exercise 114. Solve the foUowiug equations : o 3. a;(a;-10) = (6|-a:)10; {6 x + ^f = 756^ + 5x. 35 - 2 a; 5x^ + 7 __ 17_-Jj; 9 "*"5a,-2-7~ 3 7. I (2 a; - 5)2 = 94 - 24x; 3 ar^ - 4 = ^±j? . 8. -o = -o > rt ar + 7yj it' = a c^ + ??i x. 3r — n or — m 9. 2:2 ^ ^^aj_^ = ^j2:(l — wa;); 2 + 4.i'2 _. ^^(1 __^ 10. 7^ — n X ■\- m = n X (x — \)', x VB + x^ = 1 + x^. -- mn — ic ?i — aa; . .-^ 5 11. = ; X + vx^ — .3 = n — mx an — x ^x^ — 3 12 ^ I ^ ^,. 1 , 1 ^^^ • 5+^ 5_a; > 1_>/1_^ l+Vl-a;2 a:2 314 ELEMENTS OF ALGEBRA. 132. Example 1. Solve x^ — 2ax + 4ab=: 2bx. Solution. Transposing, we have x^ — 2ax + 4ab — 2bx= 0. Arrange in binomial terms and factor, and we have (x — 2 a) (x — 26) = 0. A product cannot be zero unless one of the factors is zero. Hence, the equation is satisfied if x — 2 a = 0, or x — 2 6 = ; that is, a x = 2ay or if X = 2 b. Example 2. Solve ^ x2 + f x -f- 20^ = 42f + x. Process. Clear of fractions, transpose, and unite, 3 x2 - 2 X - 133 = 0. Factor, (3 x + 19) (x - 7) = 0. Therefore, 3 x + 19 = 0, and x - 7 = 0. x = - 6^, and x =: 7. Hence, To Solve a Quadratic Equation by Factoring. Simplify the equation, with all its terms in the first member ; then place the fac- tors of the first member separately equal to zero, and solve the simple equations thus formed. Exercise 115. Solve the following equations : 1. rz;2_i0;i;=:24; a?2+2a; = 80; ^2_ iga? + 32 = 0. 2. a;2 + 10 = 13 (x + 6) ; a^ + 4x - 50 = 2 - 5 X, 3. 4x2+13^+3 = 0; Zx^ -^ 1 = -\\x - x^ ^ ^. 4. a;2-ic= 11342; 5a^+3x-4==8aj-7rz;2_2. 5. \-Zx-x^ = 2x--^x-Z\ x^-2ax-\-%x = \^a. 6. :i;3_5^2^5^ + 7^2. a^_|^ + _9_^0. x-\-Z 2x-Z 3-x 7. lla:2_iii =9^. x+ 2 x-1 x-^2 QUADRATIC EQUATIONS. 315 8. (a;-2)(a;H9a: + 20) = 0; 2x^ + Sa^^2z-S = 0. 3ar— 4 3a — 2a; 4 10. mqx^ — mnx-\-pqx — n2y = 0; x+5 = Vx+ 5-\-6. 11. {a-b)x^-(a + b)x + 2b = 0.; x^=21 + Vx^-^' 133. An aflfected quadratic equation can always be solved by the method of completing the square. This method consists in adding to both memlxjrs such an expression as will make the memlier, with all the terms containing the unknown number, a perfect square. The explanation of this methocl depends upon the principle that a trino- mial is a perfect st^uare when one of its terms is plus or minus twice the product of the scjuare roots of the other two. This process enables us to extract the square root of the member containing the unknown number, and thus form two simple equations which may be solved separately. 2x- 11 Example. Solve ^(S-x) - ^^ = i (a? - 2). Process. Clear of fractions, transpose, and unite, -4x^ + 26x= 12. Divide by -4, ar»-J^a: = -3. Add * (l^)« to lioth members, x^^^x + (J^f = - 3 + (J^y = ^. Extract the square root, x — ^z=±^. Therefore, x-^ = ^, &nd x-^ = -^. x = 6, and x = f Every affected quadratic equation may be reduced to the general form 7Ji2^ + 7ix -\- a = 0; where m, n, and a represent any numbers whatever, positive or negative, integral or fractional. Dividing both members by m for a n convenience, representing - by 6 and - by c, and transposmg, we have 316 ELEMENTS OF ALGEBRA. x^ -\- c X = — b. Add (2) to both members, x^ + cx+i^j =-^+(2)- Or, x^+cx+ (0 =|(c2-4i). c Extract the square root, x + ^ = ±^ a/c'^ — 4 6. Therefore, x + ^ = ^ ^.c^ -4 b, and x + ^ = -^ ^c^- 4 6. From which a: = - 9 + | V^'"^ " 'I ^' ^"^ ^ = ~ 2 ~ ^ V^^ ~ ^ ^- "^^^^^ -c-l-Vc2-46 values may be written in the form x — 5 • Hence, Common Method of Solving Quadratics. Reduce the equa- tion to the form x^ + c x = — b. Comflete the square of the first member by adding to each member of the equation the square of half the coefficient of x. Extract the square root of both member's, and solve the resulting simple equations. Notes : 1. * Always indicate the square of the expression to be added, in the first member. 2. Since the squared terms of the square of a binomial are ahvays positive, the coefficient of x"^ must be made + 1, if necessary, before completing the square. This may be done by multiplying or dividing both members by — 1. 3. The foregoing method is called the Italian Method, having been used by Italian mathematicians, who first introduced a knowledge of algebra into Europe. 134. It is often convenient to complete the square without first reducing the simplified equation to the form in which the coefficient of x^ is 1. Thus, 3ar-7 4a;- 10 7 Example 1. Solve — - — + ^ _^ ^ = ^ • Process. 1 = 7. 3X7-7 = + 4 X 7 - 1(1 7 + 5 " --h 2 + i-- = i. J = --i- QUADRATIC EQUATIONS. 317 Process. Clear of fractions, 6 a:2 - 1 4 a: + 30 X - 70 + 8 a:2 - 20 ar = 7 a;2 + 35 a:. Transpose and unite, 7 ar'^ — 39 a; = 70. Multiply by 7X4, 196 ar^ - 1092 x = 1960. Add (39)2, 196 x2 - 1092 x + (39)« = 1960 + 1521 = 3481. Extiaet the si^uare root, 14 a: — 39 = ± 69. Therefore, 14a: = 39 + 59, and 14a: = 39-59. x = 7, and x = - Y- Verify by putting these numbers for x in the original equation. 3 X - V - 7 4 X - V - 10 _ -1^ + -Y + 5 ~*' U - ¥ = I. i = i- When a quadratic e({uation appears in the general form »nx2 + nx + a = 0, the terms containing x may be made a complete square, without first dividing the equation by the coefficient of z*. Thus, Transpose a, mx^-{-nx = — a Multiply the equation by 4 m and add the square of n, 4m^x^ -\- 4 7nnx -\- n^ = n^ — 4a m. Extract the square root, 2 wi x 4- n = ± ^/n^ — 4am. Transpose n, 2 w x = — »» ± >^n^ ~ 4am „, . — n ± \/w* — 4am Therefore, x = . Hence, Hindoo Method of Solving Quadratics. Reduce the equa- tion to the form in x^ + n x = — a. Multiply it hy f(/icr times the eoefficient of x^, and complete tlu sqtiare hy adding to each member the square of the coefficient of x in the given equation. Extract the square root of both member s^ and solve the resulting simple equations. 318 ELEMENTS OF ALGEBRA. If the coefficient of x in tlie given equation is an even number, the square may be completed as follows : Multiply the equation by the coefficient of x^, and add to each mem- ber the square of half the coefficient ofx in the given equation. 8x 20 Example 2. Solve —7—0 "" o~ == ^* 3. "t" z o X Process. Free from fractions, (3 x) (8 x)-20(x + 2) = 6 (3 x) (x + 2). Simplify, 6x^-5Qx = 40. Multiply by 6, 36 x^ - 336 x = 240. Add (Af )2, 36 x^ - 336 X + (28)'^ = 1024. Extract the square root,* 6 a; - 28 = ± 32. Transpose, 6 a: = 28 + 32, or 28 - 32. Therefore, x = 10, or - f Verify by substituting 10 for x in the original equation. 8 X 10 20 Process. _______ ^ 6, ¥ -1 = 6, 6 = 6. Verify by substit-uting — f for a: in the original equation. 8 X - f 20 Process. _^--^ _____ = 6, -¥ 20 _ - 4 + 10 = 6, 6 = 6. Notes : 1. * We ought to write the double sign before the root of both members. Thus, ± (6 x - 28) = ± 32, tlie reason for not doing so is the same as given in Art. 131, Note. 2. The Hindoo, or Indian Method, is supposed to have been discovered by Aryabhalta, a celebrated Hindoo mathematician, and one of the first inventors of algebra. It is not only more general in form, but much better adapted to the solution of equations in which the coefficient of the square of the unknown number is not 1. 3. This method has an advantage over the common method in avoiding fractions in completing the square, and is often preferred in solving literal equa,tions. QUADRATIC EQUATIONS. 319 135. In case the coefficient of the square of the unknown, in the simplified etjuation, is a square number the square may be completed as follows : Example 1. Solve 72 a: - 54 = (20 - z)(4x + 3). Process. Simplify, 4 x^ — 5 x = 114. ""^•^ (i:Jb)' " 6)' ^ -' - ^ - + («' = Mj»- Extract the root, 2x- i = ±^. Transpose, 2 x = ^ + i,a. or | — ^. Therefore, z = 6, or — 4|. The coefficient of x^ may always be made a square number by multiplication or division. Hence, General Method of Solving Quadratics. Add to each member the square of the quotient obtained from dividing the second term by twice the square root of the first term. Tfien proceed as before. ^ , 5 3 35 Example 2. Solve —-r-, + - X + 4 X X — 2 Process. Free from fractions, 5 (x - 2) X + 3 (x + 4) (x - 2) = 35 (x + 4) x. Simplify, - 27 x^ - 1 44 x = 24. Divide by - 3, 9 x« + 48 x = - 8. (48 X \* — -= ) , or (8)«, 9 x2 + 48 X + (8)2 = 56. Extract the root, 3 x + 8 = ± 2 y/\i. Transpose, 3x = -8±2y'14. .♦. x= r-^^^ Note. The Common and Hindoo Methods of completing the square are modifications of the General Method. 320 ELEMENTS OF ALGEBRA. Exercise 116. Solve the following 1. 23a?= 120 + a:2; 42 + ^2^ 13 a;; 12 ic^ + ^ ^ 1. 2. 22a:+ 23-2;2 = 0; a:2- |rz^=i 32; 2^2 + 3 2; = 4. 3. a;+22-6 2;2=. 0; 25 ^=62:2+ 21; x2-2x = X 4. 3x'^+12l = Ux; -^i-x=--^-x^; 91 a:2- 2a; = 45. 5. 21 ^2 _|. 22 ^ + 5 = ; 9 ^2 - 143 - G a; = 0. 6. 18a;2-27ir- 26 = 0; 50 a;2 - 15 a; = 27. 7. 192;= 15-82;2; ^2+_4^^^1. ^2_l^_l3zzO. 8. 5 2;2 + 14 2; - 55 ; (2; + 1) (2 2; + 3) = 4 2:2 - 22. 9. 2(^-^)-3(2.' + 2)(.2;-3); .32^2 + 2.1 2: + | = 0. 10. 252:+22;2= 42; |(2; + 6)(2;-2) - f (62jV + -V-^)- 1 __ _1 ^ ^+16 11 _ 42: -171 ^^' IT^ 3^^~35' "T ■^¥~ 3 42; x — 6_4:X+7 '^Jl^ 2! — 2 _ ^ ^^- "9" "^ ^+~3 ~ ~T9~"' ^~^^ '^ ^^^S ~ ^^^ .0 _J__4^__L_. _i L-^1- 3-x 5 9-2 2;' 2^-3 2:+5 18 14. 15. QUADRATIC EQUATIONS. 321 4 3 4 5 3 X — 2 X X -\- ^^ X — \ x-f2 X >y2+ 3 _ 12 + 5a^ 'dx 2a;-5 _^,^ -^^ "^aj2- 5 ~ 5(0:^-5)' 2:+ 1"^ 3a:-l~*^*8- ^^ 5a;-7 a: - 5 3 a: - 1 , 10. = = = -^ zTT^ ; -^ -—=, = 1 — Ix-b 2a;-13' 4a;+7~ x -\- 1 ,^ 12a:8-lla;'-*+ 10 a; - 78 ,, , 17. 5-0 — ;, — -T-^ = l\x — h 3a:+5 3x-5 _ 135 7 21 22 3a;-5 3a; + 5~176^ x^+'^x^ '6x^-^x~ x \ 18 7 8 a; ic + 3 19. «— 1 a; + 5 a;H-l x — b' x + % 2a;+l 136. Literal Quadratic Equations. Example 1. Solve mx"^ -\-nx = — ; — -^ m x - n x^. Procesa. Transpose and factor, (m 4- n)x^— {m — n)x = — — • Multiply the equation by 4(m4- n) and add the square of (m — n)^ 4 (m + nyx^ - 4 (m2 - n^) a: + (m - n)« = (m + n)«. Extract the square root, 2(m-}-n)a:— (m — n) = ± (to + n). Transpose, 2 (to + n) a: = to — n ± (m + »») = 2 TO, or — 2 n. «« * TO n Therefore, x - , , or TO + n' TO + n « « , 2a:+l 1/1 2\ 3x4- 1 Example 2. Solve — r ( r — ~ ) = — :: — x\o aj a 21 322 ELEMENTS OF ALGEBRA. Process. Free from fractions, ax(2x+l)- (a-2b) = bx(Sx + l). Simplify and transpose, 2ax^-i- ax -3bx^~bx = a-~2b. Express the first member in two terms, (2 a - 3b) x^ + (a - b) X = a - 2b. Multiply by 4 (2 a - 3 b), 4 (2a-36)2a;2+4 (2a-36) (a-b)x = 8 a^-28ab + 24 b\ Complete the square, 4(2a-36)2a;2+4(2a-36)(a-6)2: + (a-&)2 = 9a2-30a6 + 25 62. Extract the square root, 2{2a - Zb) X + {a - b) ~ ± {3 a - bb). Transpose, 2 (2 a - 36) x = - (a - 6) ± (3 a - 5 6) = 2a-4&, or-2(2a-36. a-2b Therefore, x = ^^-^g^ , or - L 1111 Examples. Solve ^ + ^"i;-^ =-+ ^ ^ ^ 111 1 Process. Transpose, - - " = ^-+6 " j+^ Reduce each member to a common denominator, a — X X — a ax ia + b)(b + x) Free from fractions, {a-x){a-\-b){b + x)= ax(x- a). Transpose and factor, (a - X) [(a + b)(b + x) + ax'] = 0. Hence (Art. 11), a - a; = 0. .',x = a. Also, (a-\-b)(b + x) + ax = 0. Simplify and factor, ^ ^ b(a + b) b(a + b-) + (2a + b)x = 0. .'. x= - ^^^^ ■ Kote. Always express the first member of the simplified equation in two terms, the first term involving x'^, the second involving x. QUADRATIC EQUATIONS. 323 Exercise 117. Solve the following equations : -. 9 / . r\ . 7 f\ 2 . ^^^ m ax 1. x^ — (a-{-b)x + ab = 0: mx^-] m = — 7 — . ^ X a X b ^ mbx am z 2. - + - = - + -; ma^ m = r a X b X a 3. (a--6)r^-(a + 6)a;+26 = 0; a^ 3^ ^ abx=^ 21?. 7^ X ^2a^ 2 z(a — x) _ a a^ + m2_ a^'^ b^~W' %a-2x ""V ^ ~ f>, ^ — n X •\- 'p X ^ n'p ■=■ ^ \ a^ -\- 2x Vn = n. 6. a^3^ ~-2a^x+ n* -1 = 0; a^ -\- x (a - b) = ab. 7. x^-\-mx-^cx-\-Ji^x+m^x = 0. 8. ^a^x = (a^-b^ + xf; a^ {x - a)^ = 1)* (x + a)^ \a X J \ X a J 10- —i . rxQ = (« — ^)^ ; ^— ^ X — X = —CX — Z. {a-\r by ^ ' b n 11. («'~^(^J+l) ^2a?; 9a*6*2^»~6a8 63^ = 62. 19 ^^+ <^ _ ^4-6 11 1 _^ « — 6~~na; — a' a a + a; a + 2a7~~ 324 ELEMENTS OF ALGEBRA. a-x X b (a-l)^s^ -{- 2(Sa^l )x _^ lo. -r ■ " — — - ; ;; z — 1. X a — X c 4a — 1 14. ra; + ^y = 4a:2; (ax-'^=\a?x\ 137. Solution by a Formula. From the quadratic equa- tion moi?' -{■ nx — ^ a, — n± Viv^ — 4:am , ^ . *= 2Vv (^> By means of this formula the values of x, in an equation of the general form, may be written at once. Thus, Example L Solve 10ic2_ 23 a: = - 12. Process. Here, m = 10, n = — 23, and a = 12. c V .-. . .u 1 • /ix -(-23)± V(-2 3)-^- 4X 12X 10 Substitute these values m (1), x = ^ y 10 23 ±7 ~ 20 = i or f ^ r. -, 1 111 Example 2. Solve r- ; — = j- + - h b + c -\- y c y Process. Free from fractions, transpose, and factor, (h + c)y^+ (h + cyy = -hc{h-[- c). Divide by 6 + c, y^ -^ (h+c)y = — he. Here, m = 1 , n = 6 + c, and a = hc. Substitute these values m (1), x = ~ -(b + c)±(b-c) ~ 2 = — r, or — 6. Note. In substituting the student must pay particular attention to the signs of the coefficients. QUADRATIC EQUATIONS. 325 Miscellaneous Exercise 118. 1. 17 a^ f 19 a* = 1848; Sa^ - 12x + 1 = 6x-2S. 2. 5a:2 + 4^. = 273; ^x' + lx + ^ = 0, ^•^+2^=& «(^' + 3)«=|(.+ 3)^-f 4 x + 1 _ a;+2 _ 2a; + 16 x-2 '^'^'^bx'^ 5 ~^' x-l~ x + 5 x + 1 2a;+3 7 — x 7—3x 5. 16x^-6x'-l = 0; 2(2a:-l) 2(a:+l) 4-3a; a 5x_^ x-S 2 (a: + 8) ^ 3 a: + 10 3"^'4~3a^a:-5"'" a:4-4 ~ a:+l 7.--^=^; H(5^ + 36)2 = V^j(8a:2-4)^. X CL 8. -4- = - + -; lla?»4-10aa: = ±a2; a:' + - + 2 = - p-\-x p x' 'XX ,- ,2:2j2.fl,2fltJ 9. ax ■\- dx^ — a-= d\ — « H = — o H ir^ c 7?r c 10. WX2 (w^ — w^ a; _ a^ a _ mn '3m — 2a 2 4a — (3 w 11. a:^ - 2 a X = (6 - c + a) (6 — c — a). 12. x^ - (a -{- b)x = \{7n -\- n -\- a -\- h){m -^ n-a-h). ,o 1 _^ 1 a2 + a:2 ^^3 ^r - 3 a -k- X a — X a* -- or x — 6 x -\- .* 326 ELEMENTS OF ALGEBRA. 14. ahx'^ - 2x{a + h) ^/'^ = {a -hf. ' ^' "^ aW' ~ 18aH2 + 2ah a — 2h~x 5h — x 2 a — x — 19h_ a^ — 4:b^ ax + 2bx 2bx — a x ~~ ' ^^1 1 m 2nx+ n ^7 I . =: 2x^-{-x~-l 2^2— 3 a; +1 2nx — n mx^ — m 18. = ax. ax — Va^ x^ — \ ax + \o? x^ — 1 Query. What is the diflference between the meaning of "the root of an equation" and "the root of a number"? 138. Problems. The following problems lead to pure or affected quadratic equations of one unknown number. In solving such prol)- lems, the equations of conditions will have two solutions. Some- times both will fulfill the conditions of the problem ; but generally one only will be a solution. Exercise 119. 1. Find a number whose square diminished by 119 is equal to 10 times the excess of the number over 8. Solution. Let x = the number. Then, x — 8 = the excess of the number over 8. Therefore, a:^ - 119 = 10 (a: - 8). The solution of which gives, a: = 13, or — 3. Only the positive value of x is admissible. Hence, the number is 13. Note. In solving problems involving quadratics, the student •should retain only those values for results that will satisfy the conditions of the problem. PROBLEMS. 327 2. The difference of the squares of two consecutive numbers is 17. Find the numbers. 3. Find two numbers whose sum is 9 times their differ- ence, and the difference of whose squares is 81. 4. Find two numbers, such that their product is 126, and the quotient of the greater divided by the less is 3 J. 5. Divide 14 into two parts, such that tlie sum of the quotients of the greater divided by the less and of the less by the greater may be 2r^. 6. Find two numbers whose product is m, and the quotient of the greater divided by the less is n. 7. Find a number which when increased by n is equal to m times the reciprocal of the number. Find the num- ber when n = 17 and vi = 60. 8. Divide m into two parts, so that the sum of the two fractions formed by dividing each part by the other may be 71. Solve when m = 35 and n = 2^, 9. Divide a into two parts, so that n times the greater divided by the less shall equal in times the less divided by the greater. Solve when a = 14, ?i = 9, and m = 16. 10. A farmer bought some sheep for $72, and found that if he had received 6 more for the same money, he would have paid S 1 less for each. How many did he buy? 11. If a train travelled 5 miles an hour faster, it would take one hour less to travel 210 miles. Find the rate travelled and number of hours required. 328 ELEMENTS OF ALGEBRA. 12. A man travels 108 miles, and finds that he could have made the journey in 4-|- hours less had he travelled 2 miles an hour /aster. Find the rate he travelled. 13. A number is composed of two digits, the first of which exceeds the second by unity, and the number itself falls short of the sum of the squares of its digits by 26. Find the number. 14. A number consists of two digits, whose sum is 8 ; another number is obtained by reversing the digits. If the product of the two is 1855, find the numbers. 15. A vessel can be filled by two pipes, running to- gether, in 22|- minutes ; the larger pipe can fill the vessel in 24 minutes less than the smaller one. Find the time taken by each. Solution. Let x = the number of minutes it takes the larger pipe. Then, x -\- M = the number of minutes it takes the smaller pipe. - =r the part filled by the larger pipe in one minute, and ^^ = the part fi lied by the smaller pipe in one minute. r^, P 1 1 1 Therefore, - + X ' a; f 24 ~ 22f The solution of which gives, x = 36, or — 15. One pipe will fill it in 36 minutes, and the other in 1 hour. 16. A vessel can be filled by two pipes, running to- gether, in m minutes ; the larger pipe can fill the vessel in n minutes less than the smaller one. Find the time taken by each. Solve when w = 56 and w = 66. PROBLEMS. 329 17. B can do some work in 4 hours less time than A can do it, and together they can do it in 3| hours. How many hours will it take each alone to do it ? 18. A boat's crew row 7 miles down a river and back in 1 hour and 45 minutes. If the current of the river is 3 miles per hour, find the rate of rowing in still water. 19. A boat's crew row a miles down a river and back. They can row m miles an liour in still water. It took n hours longer to row against the current than the time to row with it. Find the rate of the current. Solve when a = 5, w = 6, and n = 2. 20. A uniform iron bar weighs m pounds. If it was a feet longer each foot would weigh n pounds less. Find the length and weight per foot. Solve when m = 36, a = 1, and n = |. 21. A and B agree to do some work in a certain num- ber of days. A lost m days of the time and received n dollars. B lost a days and received c dollars. Had A lost a days and B m days, the amounts received would have been equal. Find the number of days agreed on and the daily wages of each. Solve when m = 4, 71 = 18.75, a = 7, and c = 12. 22. A pei"son sold goods for vi dollars, and gained as much per cent as the goods cost him. Find the cost of the goods. Solve when m = 144. 23. By selling goods for m dollars, I lose as much per cent as the goods cost me. Find the cost of the goods. Solve when m = 24. 330 ELEMENTS OF ALGEBRA. CHAPTEE XXII. EQUATIONS WHICH MAY BE SOLVED AS QUADRATICS. 139. In the equation m {if — yy + n (y^ — z/)^ -f a = 0, suppose (y^ — VY = 'C, then mx^ + nx -l-a = 0. Similarly, y^-Sy^ — 9 = may be changed to the form a:^ — 3 a: — 9 = 0. Hence, an equation is in the quadratic form when the unknown number is found in two terms affected with two exponents, one of which is twice the other ; as, a;^ + 5 a;^ — 8 = 0. The general form for an equation in the quadratic form is, ax^'' + b^f' + c = 0; where a, h, c, and n represent any numbers whatever, positive or negative, integral or fractional. Example I. Solve a;* - 13 x^ + 36 = 0. Process. Factor, (x + 2) (x - 2) (x + 3) (ar - 3) = 0. Hence, a; + 2 = 0, a;-2 = 0, a; + 3 = 0, and x — 3 = 0. Therefore, a; = ± 2, or ± 3. Example 2. Solve 8 a;~ ^ - 15 x~^ -2 = 0. Process. Factor, (x~^ — 2) (8 a:~^ + 1) = 0. ^-i_2 = 0, ora:^ = i x = (hS^ = i ^2. Also, 8 X ^ + 1 = 0, or a;' = - 8. x= (- 8)^ = - 32. Example 3. Solve 3 a: + a:^ - 2 = 0. Process. Solve for x^. Thus, Multiply by 12 and transpose, 36 ar + 12 a:* = 24. Complete the square, 36 a; 4- 12 a:^ + 1 = 25. Extract the square root, 6 a:* + 1 = ± 5. Therefore, a;* = f , or - L Square each member, a: = ^, or 1. EQUATIONS SOLVED AS QUADRATICS. 331 Example 4. Solve 2 -^^/^^ ^ 3 ^^ _ 55 _ q Process. Since -ij/x-^ is the same as x~ ', and /y/x-*^ is the same as x~ *, this equation is in the quadratic form. Transpose and mul- tiply by 12, 36 X-* + 24 x~^ = 672. Complete the sciuare, 36 z"* + 24 x~* + 4 = 676. Extract the square root, 6 a:~ * -f 2 = ± 26. x~^ = 4, or — ^. Therefore, a:' = J, or — ^. Extract the square root, a:* = db ^, or ± \/~ ^, Raise to the 5th power,* x = ± jij, or ± V(~A)*- Hotes : 1. When the roots cannot all be obtained by completing the square, the method by factoring should be used. Thus, in solving a;* + 7 a:* — 8 =: 0, by completing the square, we find but two values for x, re = 1, or — 2. Fac- toring the first member, we have (x + 2) (a* — 2 x + 4) (x - 1 ) {x^ + x-\-l) = 0. Hence, a; + 2 = 0, a^ _ 2 a: + 4 = 0, a; - 1 = 0, and x^ -f a: + 1 = 0. Solv- ing these equations, a: = — 2, 1 ± V^^, 1, and 2. • In solving equations of the form a;» = a, first extract the wth root, and then raise to the »ith power. In practice this is the same as affecting the •quation by the exponent — . Thus, a: = a« . m Example 5. Solve a a:^" + 2» jr" = — c. Process. Multiply by 4 a, 4 a* x^'* + 4a6ar"=: — 4ac. Complete the square, 4a2j:2" + 4abx'* + b^ = -4ac+b^. Extract the square root, 2 a a* + 6 = ± yb^ — 4ac. Transpose b and divide by 2 a, af = — ^- — • Extract the -th root, x = [±VE^ILz*]" (i) Example 6. Solve Ax* - 37 x' + 9 = 0. 332 ELEMENTS OF ALGEBRA. Process. Here, a = 4, 6 = — 37, c = 9, and n = 2. Substitute these values in(i), x = [ ^ V(-37)^-4X4X -9-(-37)f ^^' L 2X4 J r ± 35 + 37 1^ = ± 3, or ± i Exercise 120. Solve the following equations : 1. 0^4-14^2^-40; a;io + 312^5 = 32; x^-7x^ = S. 2. 2^(19 + :i^3):^ 216; 2^2 + ^ ^.^ ,,2 + ^2 3. 16(':c2+ i^ = 257; ^3 +14^^1107. 4. 5 a:* + \/x = 22 ; 'v/^ + -|- = SJ. 2 V 2- 5. ^-t + 7 ^t = 44; 3 a;3 + 42 x^ = 3321. 6. x^ + x^= 756 ; 3 V^^ _ 4 ^.^ = 7, ^ . .. _2 . 2 Vx ' xi 7. 2V^ + ^^ = 5; 122:-t + f = 4 + ¥- a 3 ^t - a;-| + 2 = ; 2 a;-5 + 61 a^-t - 96 = 0. 9. x-^ + ax-i = 2a^; x-^-2x-^ = S', x-^ + V^=(^ 10. rr*" - |2;2» - || = ; 3 ict'* + 4 xl"" = 4. 11. a:" + 13 o:''* = 14 ; 3 :r '"^ - 26 x ^'" = -16 EQUATIONS SOLVED AS QUADRATICS. 333 140. ?>iuatioijs may frequently be put in the quadratic form by grouping the terms containing the unknown number, so that the exponent of one group shall be twice the exponent of the other group, and then solved for the polynomial. Thus, Example 1. Solve a; - 3 x* - 4 Va: - 3 x* - 1 = - 2. The aquation may be put in the quadratic form if we reganl Vx — 3 a:* — 1 as the unknown number. Thus, Process. Add — 1 to each member, x-3a;*-l+4Vx-3ar*-l=-3. Put Vx -3x^-1 = y, y^ + 4y = -3. Therefore, y = 3, or 1. Hence, Vx - 3 x* - 1 = 3, or 1. Squaring, ' x - 3 x* — 1 = 9, or 1. Complete the square, x - 3 x* + } = ^, or Jf . Solving these equations for the values of x, we find x = 25, or 4, 13±3v^ and X = ^ — — Hote 1. In solving equations of this fonn we must group the terms so that the expression outside of the radical, in the first member, is the same or a mul- tiple of the expression under the radical sign. Example 2. Solve x* - 6 ax« + 7 a" x^ 4- 6 a»x = 24 a*. Process. Add 2a«x^ x*-6ax^+9a^x^-^6a*x = 24 a* -f 2a«x«. Transpose 2 a«x2, x*- 6 a x«+ 9 a^ x2+ 6 a«x - 2 a^x^ = 24 a*. Group and factor the terms, (x2 - 3 ax)2 - 2 a2 (x2 - 3 ax) = 24 a*. Regard x*— 3 ax as the unknown number, and complete the square, (x2 - 3 a x)a - 2 a« (x2 - 3 a x) -f- a* = 25 a*. Extract the square root, (x'^ — 3 a x) — a^ = ± 5 a* Therefore, x« - 3 a x = 6 a", or ~ 4 a«. Complete the square and solve, x = 2 (3 ± \/33), 334 ELEMENTS OF ALGEBRA. Note 2. Form a perfect square with xi and —6a x^. The third terra of the square is the square of the quotient obtained by dividing 6ax^ by twice the square root of x*. Example 3. Solve a:2 + 4x — 4a:r-i-fx-2 = ^. Process. Use positive exponents, rearrange terms, and factor, a=^ + ^, + 4(x-i)=|. Regard x as the unknown number, and subtract 2 from both sides, „ ^ 1 . f 1\ 11 Factor, and complete the square, (.-iy+4(.-^)+4=^. Extract the square root, a: — - + 2 = ±f. Therefore, x — - = — ^, ot — ^. X Free from fractions, ^ ar^ — 1 = — ^ a:, or — ^ x. Complete the square and solve, x = ^ (— 1 ± y'S?) , x = ^(-ll±^/T57). Note 3. Form a perfect square with x^ for the first term and — for the third. The middle term will be twice the product of their square roots taken with a negative sign. A Biquadratic Equation is an equation of the fourth de- gree. Biquadratic means twice squared, and hence the fourth power. If a biquadratic is in the form, x^+2mx^+ (m2 + 2 n) a:^ + 2 mnx = a (ii) the first member becomes a perfect square by Adding n^, or the square of the quotient obtained by dividing the coefficient of x by the coefficient of x^. EQUATIONS SOLVED AS QUADRATICS. 335 Thus, extracting the stiuare root of the fii-st member, X* i- 2mx^ + {inr +'2,n)x^ -\- 2mnx | x'^ + mx -{- n X* 2x^-\- mx I 2mx^+ (m^ + 2n)x^-\-2mnx 2mx*+{m^ )x^ 2x^ + 2mx-\-n\ + ( 2n)x^ + 2mnx + ( 2n)x^+ 2mnx + n^ — n*. Hence, the equation may be written, {x^ -\-mx + n)2 - n^ = a, or (z" + w z + «)« = a + n^ (iii) Example 4. Solve x* - 10 «« + 35 z^ _ 50 x = 1 1. Process. Here, 2m = — 10, 2 win = —50. .-. m = — 5 and n = 5. Since m^ + 2n = 35, the equation has the form of (ii). Add 25 ; or put w = — 5, n = 5, and a = 1 1 in (iii), (z2 - 5 z + 5)2 =r 36. Extract the square ^oot, z* — 5z + 5 = ±6. Therefore, z^ - 5 z = 1, or - 11. 5 ± a/29 Complete the square and solve, z = ^ — , 5± a/^Hq ^ = 2—- Kote 4. After adding the value for n* the first member may be factored by substituting the values for m and n in (iii). Exercise 121. Solve the following equations : 1. (3^ + x-'2)^-lS(2^ + x-2) + Z6 = 0. 2. a^» 4- 2 a; + 6 Va^+2x+5 = 11. 3. 2:2 + 24 = 12 V^-qrf. 2a: + 17 = 9 V2 a; - 1. 4. a:2 - a: + 5 (2a:2 - 5 a; + 6)i = J (3 a: + 33), 336 ELEMENTS OF ALGEBRA. ^ (a;2 _^ ^,. ^ 6)i ^ 20 - |(a;2 + a; + 6)^ 7. (« + l)%4(«+!) = I2. 8. (a;2 - 5 a;)2 - 8 (ic2 _ 5 ^) ^ 43^ 9. 9 ^ - 3 ^2 _!_ 4 (^2 _ 3 ^ + 5)^ ^ il^ ■«(-9"-!(-^)-S- 11. (3a:2-10^+ 5)2-8(3^- 10a^+ 5) = 9. 13. :i:4 + 6^^ + 5:z;2- 12a;=12. 14. x^-Qa^-2^x^+ 114 2; = 80. 15. ^4 + 2 2^ - 25 a;2 _ 26 a; + 120 = 0. 16. 2:4-8 .>;3 + 10 2:2 + 24 a: + 5 = 0. 17. a:4 + 8 2:3 + 2 a:2 - I a: = |_ 18. (^3 _ 16)1 _ 3 (2^3 - 16)i = 4. 19. ^-Y^^-Vx-x-^^A.; 2;2+3:^-32:-i + 2r2 = ^. EQUATIONS SOLVED AS QUADRATICS. 337 141. Equations Containing Radicals may be Solved. Thus, Example 1. Solve x - ^3* + 2 x + 12 + 2 = 0. Process. Transpose, x + 2- ^x» + 2x -f- 12. Raise both members to the third power, a:» + 6 x-» -f 12 X + 8 = a:* + 2 X -h 12. Transpose and simplify, 3 a:"^ + 5 x — 2 = 0. Factor and solve, x = ^, or — 2. Verify by putting these numbers for x in the original equation. Process, x = J. x = 2 - ^-8-4+12 + 2 = 0, -2-0 + 2 = 0, = 0, 1 1 i - ^tjV + f + 1^ + 2 = 0, i - i + 2 = 0, = 0. Example 2. Solve Vx* + 1 ^/x'^ - 1 ^/J^ - 1 Process. Multiply by /^/x* — 1, Vx2 - I + yx« +1 = 1. Square, x^ - 1 + 2 ^/x'^~^\ + x« + 1 = 1. Transpose and simplify, 2 ^/x* — 1 = 1 — 2 x^. Square, 4a:*-4 = l-4x« + 4x*. Simplify, x« = f . Extract the square root, x = i J y'S. Exercise 122. Solve the following equations : 1. 3 V^+6 + 2 = a;+VT+6; a:+ \/iT~2 = 10. 2. aj 4- 16 - 7 v./- + 16 =10-4 Vx + 16. 3. 2a:+ V4a; + 8 = J; V4a;+ 17+ V^l » 4. 338 ELEMENTS OF ALGEBRA. 4. 2'v/3a; + 7 = 9-V2i^-3 - V4:X + 2 4 - Vi 4A/ic Vx 12 5x- 9 _ \/5^-3 Vic+ 12' V5^+3 2 44-2; 5. V^ + ^ = y ; -7= — ;==! + Vic + 12 V a; + 3 6. Vi^-2\/^-=2;; 1^64 -{-2x^-Sx- ,, V4: -\- X H, » A/ . * A/ o /o— 3.a; — 1 . , Vi^a; — 1 + ^ = a;. X + V'2 — x^ X — V2 — x^ ^ V7 y2 + 4 + 2 a/3 7/ - 1 ^ m- Vrn^ - y^ a/7 2/^ + 4 - 2 A/3y- 1 ' ^^^ + A/m2 - 2/2 10. ^a2 + 2a^2_2aaj = -^^iL; A/6a:-a;3= "^ 11. Va + .^' A/a; a^-&2 V^ + & a/.^: + 9 3a/^-3.8 a/^ 2; + & ^ ' Vx 9 — Vic ^^ /— -- / 12a 6 + SVx 4 12. Va + X -{■ ya — x= ; 7— r — 7=^ = "7= 5 A/a + a; 4 + Y^,^ Va; 13. V^f^+Jj - Vy^^ = V2y; 2rr+3A/^=27. , , a; + a/S 2:2-2; 12 + 8 a/S 14 ^ — — — • X — -z X- Vx 4 ' x-b o_^ + ^'^^. a; _ A/a; — 12 2; '4 2: — 18 16. A/2;^ + A/2r^ = 6 a/5; Vx — a ■\- "sJx -^ a = ^/'^ THEORY OF QUADRATIC EQUATIONS. 339 THEORY OF QUADRATIC EQUATIONS. 142. Representing the roots of mx^ + nx = ~ a by r and r^, we have (Art. 134), — n + \/n^ — 4 a m ^ — n - \^n^ — 4am '1- 'Zm n ' + '. = -» (i) a rr, — — *■ m (ii) Adding, Multiplying, Hence, if a quadrntic appears in the form, mx^ + nx = — a^ I. The sum of the roots is equal to the quotient ^ with its sign changed, obtained by dividing the coefficient ofx by the coefficient ofx^. II. The product of the roots is equal to the second member, with its sign changed, divided by the coefficient of x^. By means of (i) and (ii) the ori«,Mnal equation becomes, m x* — wi (r -f r,) a: 4- m r Tj = (1) Factor, m(x-r)(x - r,) = (2) If m = 1, x^-\-nx = — a x^- (r + rj) X + rrj = (iii) ix-r)(x-r,) = (3) If the roots of a quadratic equation be given, by means of (iii) we can readily form the equation. Example 1. Form the equation whose roots are ^, —\. Process. Here, r = ^ and r^ = -~ \. Substitute these values in (iii), x^- (^-\)x + (\) (-\) = 0. Simplify, 8 x« - 2 x - 1 = 0. Example 2. Find the sum and the product of the roots of 8x«+3x-5 = 0. Process. Here, m = 8, n = 3, and a = — 6. Substitute in (i) and (ii), r + r j = — { and rrj = — j. 340 ELEMENTS OF ALGEBRA. Exercise 123. Find the sum and product of the roots of : 1. 2:2 + 8^ = 9; 12 a:'^ - 187 ^" + 588 = 0. 2. 20x-^ = 5-5x-^l a^-6x+9 = 9x. 3. 32:2 + 5 = 0; ^24.^2:=^^. a^-l5x = S. . „ 2mn^x mn 207, 2 . 1.2 n 4. x^ = ; x^ — 2 X — 0? -\- h^^ = 0. m — n m — n Form the equations whose roots are : 5. 7,-3; |,-|; 5, -3; ± V=^; 2- V3, 2 + V3. 6. 0,-5; 7 + 2A/5, 7-2a/5; 1 + V2, 1 - V2. 7. 7/1 (m + 1), 1 — //I ; — , ; 1- , — a. ^ ^ n m a — h 8. - w + 2 \/2 /I, - w - 2 V2 71, > -^— 143. A Root is said to be a Surd when it can be found only approximately ; as, a; = =t ^^, Real Roots are values of the unknown numbers that can be found either exactly or approximately. Imaginary Roots are values of the unknown numbers that cannot be found exactly or approximately; as, x = ± V^^. Character of Roots. For brevity, represent the roots of the equation mx^-^nx + a- by r and r^, then, r= TJ}L , 2m _ — n — \/7i^ —4 am ^^ 2m "' THEORY OF QUADRATIC EQUATIONS. 341 It is seen that the two roots have the same expression, y/n^—Aam. If n^ is greater than 4 am, n* — 4 a m will be positive^ and \/n* -4am can be found exactly or approximately. If n is positive, r^ is numerically greater than r ; if n is negative, r is numerically greater than Tj. Hence, I. Condition for Eeal and Different Ebots. n* - 4 a m, positive. nitiatration. 3a:2-2x + | = 0. Here, wi = 3, n = — 2, and a = |. n2-4am= (-2)2 -4X^X3=4-1 = {. Therefore, the roots are real and different. Evidently both roots will be rational or both surds according as n* — 4 a m is, or is not, a perfect square. Hence, 11/ Condition for a Rational or a Surd Root, n^- 4am, a square number; or, /y/n- — 4a7/t, a surd. lUustrationB. (1) z* - 3 x - 4 = ; (2) 8 x^ + 5 x - J = 0. (1) Here, m = I, n = - 3, and a = — 4. n2 - 4 am = (- 3)* - 4 X - 4 X 1 = 9 -H 16 = 25. Therefore, the roots are real and rational, and dilferent. (2) Here, m = 8, « = 5, and a — — \. Vw* -4am = \/25 + 8 = \/33- Therefore, the roots are real and surds, and different. If n* is less than 4am, n'— 4am will be negative, and \/«*— 4am will represent the even root of a negative number. Hence, III. Condition for Imaginary Roots. n«-4am, negative, Uluatration. 2x«-3x + 2 = 0. Here, m — 2, n = — 3, and a = 2. n«-4am=(-3)« -4X2X2^9- 16 = -7. Therefore, the root.<< are both iniM^'inary. If n* = 4am, n*-4«m = 0, and the roots will be real and equals and have the same sign, but opposite to that of n. Hence, 342 ELEMENTS OF ALGEBRA. IV. Condition for Equal Roots, n^ ~4am = 0. Illustration. 4x2 — 12a; + 9 = 0. Here, m = 4, n = — 12, and a = 9. 71^ -4 am =144- 144 = 0. Therefore, the roots are real and equal. If a m is positive, for real roots, n^ — 4am will be positive and less than n^, since ^n^ — 4am will be less than n. If a 771 is negative, ^n^ — 4 am will be greater than w, since n^ — 4 a m will be greater than n^. Hence, V. Condition for Signs. // a m is positive, real roots have the same sign but opposite to that of n. If am. is negative, the roots have opposite signs. Illustrations. (1) 2x2-10a;+12 = 0; (2) 2x^-5x-3\^ = 0. (1) Here, m = 2, n = - 10, and a= 12. n2 - 4 a m = 100 - 96 = 4. Therefore, the roots are rational and positive, and different. (2) Here, m = 3, n = — 5, and a = — S^J. n2 -4 am = 25 + 47 = 72. The roots are surds and have opposite signs, and different. Exercise 124, Determine by inspection the character of the roots of : 1. 5x'^-x = 3; 7x^+2x=-^: Ux^-x^-^, 2. 4^2+ 52a; = 87; 3 x^ + 4a' + 4 = 0. 3. 6-llx-9x'^ = 0; 9a = 3 + 4la;2 4. 10x+S:^ = -3x^- lx'^-^x + l=0. 5. 3a:2-2a:+3=:0; 4:X^-3x-5 = 0. THEORY OF QUADRATIC EQUATIONS. 343 8. 6ar2+ 5 2:-21 = 0; 13 ar^ + 56 a; - 605 == 0. 9. 9 ar^ - 30 a; + 41 = ; 40 o,^ - 100 a; - 360 = 0. Query. How many roots can a quadratic equation have ? Why ? Miscellaneous Exercise 125. Solve the following equations : 1. a:t + 7J = 44; x-^-2x-^ = S; 3s^-:^-2 = 0, 2. k/t-^- + J- = 2X- 1 + 8 a:* -h 9 '>y^ = 0. ▼ i — X ^ X 3. —== — 2 V2 a: = 59 ; ., ^ , ._ + — = = 3 Va;. ^/2x 1 + 5 Va; Vx 4. a;*"-2a:3« + ^ = 6. 3^-2o(^ + x = a. 5. a^ + -^a:3_39^ = 81. a^ _ 2 a^ + a; = 380. 6. 108a:*= 180^8- 20aj- Sla^H- 7. 7. a:*-10a:2 + 35 a:^ _ 50^; = - 24. 8. {x - rt)f + 2 v^ (a; ~ rt)i - 3 7i = 0. 9. a:t ~ 4 a:f + r- J + 4 a:-f = - {. 344 ELEMENTS OF ALGEBRA. V y -\- 2 a — ^y — 2 a _ y x -\- ^x _ o^ — x ^y-2a + V2/ + 2a ~" 2 a' x - '^x 4 4:r" 11. 3a;"'^a;"-^:=:4; V'6:r+l + K2: + 4+ V^^+l = 2. 12. a; V5 + a/2 a' 4- 2 = V^ + :^; a: - 1 = 2 + — - • 13. 2 Vi + 2 :z^-i = 5 ; 6 Vi = 5 :c- 2 - 13. 14. x^ + 2 m^x-^ = ^m; oT^ + 2 = ^ ,"^ • X- 3 + 5 15. ^^2;+ V2^^=T-Kx- V2^T = I v/ — ^^^=. ic + Va^ - 1 a: - V:r2 - 1 16. ^ ^ ^ -^ - :: ^ = 8 ^' Va:^ - 3 ^^ + 2. a; — Vx^ — 1 2: + Va;2 — 1 17. — ^^ — ^~^, ^^-— = & ; ic3 + a; A/a; - 72 = 0. ft + 2; 18. State the conditions that will make the roots of x^ + Ax + B = 0: (i) surds; (ii) real; (iii) imaginary; (iv) equal; (v) have same signs; (vi) have opposite signs; (vii) equal in value but opposite in sign. 19. Find a number such that if its nth root be increased by one half of its ^th root, the sum shall be a. Solve when n = 2 and a = 5. 20. Find a number sucli that if its nth power be dimin- 2 a . ' ished by the - th root of the th part of it, the remainder -' n c ^ shall be m. Solve when m = 144, n = 2, a = 27, and c = 5. SIMULTANEOUS QUADRATIC EQUATIONS. 345 CHAPTER XXIII. SIMULTANEOUS QUADRATIC EQUATIONS. 144. Only certain forms of quadratic equations involving two unknown numbers can be solved. Thus, Example 1. Solve the equations : ^ ^ ^„ ^~ « « ^. ^2 10 - 1/ ProcesB. From (1), x= ^ (3) /l()_y\2 /10-w\ Substitute in (2), 2 ( — ^-^ j - ( "^ ) 2/ + 3 3^« = 54. Simplify and factor, (y — 4) (4y + 1) = 0. Therefore, y = 4, or — ^. Substitute in (3), ar = 3, or 5^. Hence, When one of the Equations is of the First Degree. Solve by substitutiou. The Degree of a term is the mimber of literal factors involved, and is always equal to the sum of their ex- ponents. Each literal factor is called a Dimension. Thus, 3 xy is of the second degree, and has two dimensions. 5 x«j/* is of the fjih degree, and has Jive dimensions. « o 1 .1- *• 5l83xy + 72x+36y = 88 (1) Examples. Solve theequations : | ^^^^^^^^^^3^^ ^ g^ ^^^ 80-36y .^x Process. From (2), a; - Y77— jTgo ^' ^ Substitute in (1), 183(80y-36y«) 72(8<l-36y) , _„ _ ^^ 177y + 60 + 177 y 4-60 + ^^ ^ " ^^^ Simplify, 9 y* + 57 y - 20 = 0. Complete the square and solve, y = J» or — 6f . Substitute in (3), ar = ^, or - f 346 ELEMENTS OF ALGEBRA. Example 3. Solve the equations . | 6 x^ - x - 3 3/ = 5 (1) ^ lx^ + x-y=l (2) Process. From (1), y = :: (3) Substitute in (2) and simplify, 3a:2-4a:-2 = 0. _ Complete the square and solve, x = :z ^ , . . , . 11 ±7 ViO ,, Substitute m (3), y — ^ . Hence, When each Equation Contains only one Second Degree Term, and that Term Consists of the Same Product or Square of the Unknown Numbers. Solve by substitution. Exercise 126. Solve the following equations : C^xy = 50. C2x + y = 22. 2. |3.;j7/+6^-2z/ = 4. g {x-y = ^. \4.xy — X + ^y =^l. ' \xy=126. ^ (x + xy = 24. ^Q r 2^3 -7/ = 218. ' \xy + y = 21. ' \x~y = 2. 4 f2y2 + y:=28. ^^ fx--y=:4. ^ (15 + y = x. ^2 (x + 3y=16. \xy = 2ij^. ' \3x^-h2xy-y^ = -12 Q^ (xy+Qx + 7y = 66. ^3 {frJ^ + y2='iS5. \Sxy + 2x+5v = 70. ' lx-v=3. SIMULTANEOUS QUADRATIC EQUATIONS. 347 15/2^ + ^ = 9. .Q (x-y = 4. 145. All equation containing two unknown numbers is symmetrical when the unknown numbers can change places without changing the equation ; q^q, 'i a? — 4: x y ■\- Z 1/ = 2 \ ic* 4- b 3^y -\- 5 xij'^ + y^ = — 5 x^ ?/. Example 1. Solve the equations : ( x^ + ,/ = 89 (1) Uy = 40 (2) Process. Add (1) to twice (2), Siibtract twice (2) from (1), Extract the square root 0! (3), Extract the square root of (4), x2 + 2xi/-f 2/2^ 169 (3) x2-2xy + y2-9 (4) x + y = ± 13. X - 1/ = ± 3. We now have to solve the four pairs of simultaneous equations, ar-f t/=13> x + y= 13 > x -f y X- y= 3r x-y = -zy x-y r^--13) x+y = -137 = 3 y x-y = - 3>' There are four pairs of values, two of which are given by x = Jb 8, '/ = ± 5, and the other two by x = ± 5, y = ± 8, in which the upper signs are to be taken together, and the lower signs are to be taken together. Kotes : 1 . If the second members of two simple equations have the sign ± , we will have six simultaneous .simple equations to consider. 2. The above equations may be solved as in Art. 144, but the symmetrical nwtliod is more simple. K\.\Mi'LE 2. Solve the equations: \ „ ^~ „ _, \J. ^ lx^-xy + y^-2l (2) Process. Divide (1) by (2), x + ?/ = 6 (3) Square (3), x^ + 2xy-\-y^ = 36 (4) Subtract (2) from (4), 3xy= lb, or xy = b (6) Subtract (5) from (2). x^ - 2 x.V + y2 = 16. Extract the square root, x — y = ± 4 (6) Add (3) and (6) and divide the result by 2, x = 5, or 1, Subtract (6) from (3) and divide the result by 2, y = 1, or 5. 348 ELEMENTS OE ALGEBRA. Example 3. Solve the equations . 5 ^'^ + 2/^ - ^ - 2/ = "8 (1) ^ ixy + x + y = 2^ (2) Process. Add (1) to twice (2), x'^+2xy + y'^+x + ij = 156. Factor, {x + yy + (x + y) — 156. Regard a; + ^ as the unknown number, complete the square, and solve, * a; + ?/ = 12, or- 13 (3) Subtract twice (2) from (1), factor, and transpose ^(x + y), {X - 3,)2 = 3 (X + y-) (4) From (3) and (4), {x - yY = 36, or - 39. Therefore, x - y = ± 6, ov ± y.1~39 (5) - 13 ± V-39 Add (5) and (3), etc., a; = 9, or 3, and ^ Subtract (5) from (3), etc., y = 3, or 9, and -^ • Example 4. Solve the equations : < ^ ^ ^ ^ lx + y = 4 (2) Process. Raise (2) to the fourth power, x*-\-4x^y + ex^y^+ 4xy^ + y^=256 (3) Subtract (1) from (3), etc., 2x»y + 3x^y^+2xy^- 87 (4) Square (2) and multiply the result by 2 xy, 2x»y-h4x^y^-^2xy» = 32xy (5) Subtract (4) from (5), etc., x^y^- 32xy = - 87. Regard xy a.^ the unknown number, complete the square, and solve, xy = 29, OT 3. We now have the two pairs of equations to solve, X -^y= 4\ X + y= 4) xy=29) ' _xy = 3} ^ ar = 2 ± 5 \/^> From the first, A "^ ^ l2^^2T5V-l. (0; = 3, or 1. ' \y=\, or 3. From the second, ^ When the Equations are Sjrmmetrical Combine them in such a manner as to remove the highest powers of x and y. SIMULTANEOUS QUADRATIC EQUATIONS. 349 Exercise 127. Solve the followiug equations : ' \x + 7j = n. ' \x^-xy + y^=2l. ^ ix^ + x + i/+i/=lS. g (2^+x^y^+y^ = 9Sl. ' \xy = 6. ' \x^ + X y -\- i/ = 49. ^ (x^y^+2x+2y=50. ^Q (x^-xy + y^=U. \xy + x + y = 2d. ' \x + y=U. g (2^ + yi = 52. j^ ra^4-a^»v/2 + /=133. \ X + y + xy = 34. ' \ o^ + x y + y^ = 19. 6. <^ 2:2 + 2r^ - 900 • 12. ^ a?» + 2r* ~ ^ * l«y = 30. U + y = 8. 146. An algebraic expression is said to be honwgeneovs when all its terms are of the same degree. Thus, 9 x*<> -f 3 X y* — 8 r* y* is homogeneous, for each term is of the 10th degree and has ten dimensions. Example 1. Solve the equations H « « « „ « « \^i Process. Let y — vx, .iiifl substitute in both equations. From (I), 6x2-f- 2t'«z«-6t;x«= 12. 12 Therefore, ^'= 6-5»; + 2 »;« ^^^ From (2), 3x« + 2yx2 = 3 r^x^* - 3. Therefore, ^' = 3t;«-2t;-3 ^"^^ 12 3 Equate (3) and (4). e-5v + 2t^ = 3ra-2t;-3 Simplify and solve for », r = }, or - 1. 350 ELEMENTS OF ALGEBRA. Substitute y = | 12 = 4. 6-5 X 1 + 2(1)' Substitute v = — ^ in (3), 12 25 "^ 6-5X-| + 2(-f)2-3l' .-. x = i /j V31. 3/ = -f^-T^\V31. Notes : 1. In finding the last values of x and y, it will be observed that ± values of x gives respectively — and + values of y. This indicates that the equations can be satisfied only by making y ^ — ^^ V2>1, when x^-\- ^^^ V'6\ ; and when x =. — ^^ V31, y must be + ^f VZl. 2. The sign T denotes precedence of the negative value. When each Equation is of the Second Degree and Homo- geneous. Substitute v x tor y in botli equations. Exercise 128. Solve the following equations : ^ ^ x^ + xy = lb. n ( x^— o xy + y'^ = — 1. ' \y^ + xy= 10. ' \Sx^-xy + Sy'^ = lS. 2 (x''-xy = 24:. ^ (2x^-5xy+3y'^=l. \ X y — y^ — 8. * \ 3 x^—5 xy+'2y'^ = 4=. = 21. 18. 2 (x^ + 4:Xy=lS3. g (x^~2xy ' \4:xy + 16/ = 228. * I xy + y^ = ^ (2x^+3xy=26. ^ (x^+3xy = {Sy'^+2xy= 39. ' \xy + 4:y^ = ^ ( 4^2_^to/+4?/2=13. . ^ ( x^ + xy + 2 y^ ' I 8x'^-12xy+Sy^=ll. [2x^+2x1/ + y^ = 54. 115. 74. ■12^?/+8/=ll. " 12^24.2^^ + ^=73. Queries. What is a homogeneous equation % Into what forms may simultaneous quadratic equations, M'hich can be solved, be grouped % What is the degree of the equation arising from eliminat- ing one unknown number from two equations, each of the second degree ? Prove it. How may such equations be solved .^ SIMULTANEOUS QUADRATIC EQUATIONS. 351 Note. In solving the following equations the student is cautioned not to work at random, but to study the equations until lie sees how they may be combined in oixler to produce sinjple equations, and tlien perform the opera- tions thus suggested. Usually the operations of addition, subtraction, multi- plication, division, or factoring will effect a simplification of the equations. Miscellaneous Exercise 129. Solve the following equations : \xy= 15. 11. ^2^+:\u.^y-\-:\xy^-\-2y^ = {). U2-r.t// + y2^1-x'^/. 2 /^-3' = 3- 12. {.ly-\- .I25x = y — x. I y — .0 X = .7o X y — o X. ^■\x-.>,= l. 13. f .Sx + .l2by = ox-y. 1 3 a: -\-y = —2.25xy. <x> + y' = 2U. ^ \x + y = 22. 14. 1 a:* -f 7/4 = 706. \x + y = 2. (a? + i/ = 7i. { xy = o5. 15. {x + y + x^-\-y^=\S. 1 xy = 6. ^- j(y_i)^.2_3^^2. 16. {4(x + y)^3xy. 1 X + y -\- x^-\-i/= 26. 1 0^-7/2= 175. 17. (a^+ ;/ ,, 337. \x-^y = 7. (.,5+ 5 7/2 = 6 a;. ix^ + xy-\-x=\4. \x^-5y^ = 4:xy. ^^' \ y^ -{- xy + y = 2S. ^ r 2:2 4. ^y^ 140. ^^^ {2^-yS = 20S. ' \y'^ + xy = 06. " I xy(x — y) =z48. 10. i^-^r!- 20. (-?/ + =^y = 12- xy-y' = 4. ■ \y + sc>y=lS. 352 21. ELEMENTS OF ALGEBRA. 22. ^ -\- if' ^=^'^xy. X + y = b. 2xy +12 = Zx\ 6 xy -{- 12 = a;^ 2 3 23. ^ ^ y lxy = 2. x-\- y ,./: ^4. 24. 25. 26. 27. 29. 30. 31. 32. l-\- xy I - xy 2^4 + a;V + y/* = 7371. x^ — xy -\-f = 63. x^ + y^ = 641. 0^2/(^2 + 2/^) =- 290. x^ -\- 3 a; ?/ + ;?/ = 19. a;2 + ;^2 ^ lo; a;2 — 3 xy + ;/^.= — 5. 3^2_5^y+3y2^9 y^ — x^ = a^. y — X = a. x^-\-y^ = Ux^7/. X + y = a. 96 — x^y^ = 4:xy. X + y = 6. x^ — y/ = 56. x^^- xy \f-^ 28, 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. y'^ — xy = a? ■{■ W'. xy — x'^ = 2 ah, x^y — y — 21. x^ y — X y — 6. ^ 2 x^ 480 2/2 + "^ - "49" • (^a;2 + 2/^ = 65. 1 1 a; + ?/ X y 6 a; + V 5 / x -\- y -[- I xP'y'^ + 5 a;?/ = 84. a; + ?/ = 8. ■'« + y + Va: + y = 12. x^ + f^ 41. a; + ?/ + ^/x + y = 12. x^ + y'^ = 189. a; + Vxy + ?/ =: 19. x^ + xy + y^ = 133. a/^ + Vy = 4. V^3 -^ ^f = 28. a:?^ 4- 2/5 = 6. a:l + 7/1 = 126. ?/2 — a;2 = 4 a &. xy = a^ — h'^. 2x?- ?yxy + ?y2 = 4. 2a;2/- 3 2/2-:i;2^-9. SIMULTANEOUS QUADRATIC EQUATIONS. 353 48. <i^ 2^ La; y X -^ y xj-_y _ 10 49. -{ x — y X + y~ 'S ' y + / = 45. X •\- y ^ x — y 50. 'ia-^b~a — b xy = ab. 51. The sum of the squares of the digits composing a number of two places of figures is 25, and the product of the digits is 12. Find the number. 52. There are two numbers whose sum, multiplied by the greater, gives 144, and whose difference, multiplied by the less, gives 14. Find the numbers. 53. The sum of the squares of two numbers is a, and the difference of their squares is b. Find the numbers. Solve for a = 170 and 6 = 72. 54. A number divided by the product of its two digits gives the quotient 2 ; and if 27 be added to the number, the digits are reversed. Find the number. 55. The sum of two numbers is a, and the sum of their 4th powers is b. Find the number. Solve for a = 4 and b = 82. 56. The difference of two numbers is a, and the differ- ence of their cubes is 7 a' Find the numbers. 57. The difference of two numbers is 3, and the difiTer- ence of their 5th powers is 3093. Find the numbers. 354 ELEMENTS OF ALGEBRA. 58. A number consisting of two digits has one decimal place ; the difference of the squares of the digits is 20, and if the digits be reversed, the sum of the two numbers is 11. Find the number. 59. Find three numbers whose sum is 38, such that the difference of the first and second shall exceed the difference of the second and third by 7, and the sum of whose squares is 634. 60. The small wheel of a bicycle makes 135 revolutions more than the larger wheel in a distance of 260 yards ; if the circumference of each were one foot more, the small wheel would make 27 revolutions more than the large wheel in a distance of 70 yards. Find the number of feet in the circumference of each wheel. 61. Find two numbers such that their difference shall be a, and the product of their n\k\. roots c. Solve for a = 4, c = 2, and n = h. 62. Find a fraction such if the numerator be increased and the denominator diminished by 2, the result will be its reciprocal; while if the numerator be diminished and the denominator increased by 2, the result will be \% less than its reciprocal. 63. A principal of $10,000 amounts, with simple in- terest, to $14,200 after a certain number of years. Had the rate of interest been 1 % higher and the time 1 year longer, it would have amounted to $15,600. Find the time and rate. 64. A sum of money at interest amounted at the end of the year to $10,920. If the rate of interest had been 1 % less, and the principal $100 more, the amount would have been the same. Find the principal and rate of interest. INDETERMINATE EQUATIONS. 355 CHAPTER XXIV. INDETERMINATE EQUATIONS. 147. Simple Indeterminate Equations are equations of the first degree that admit uf aii unlimited number of solutions. Thus, in 3x-2y = 2, if y = 2, x = 2; if 3/ = 3, a; = 2f ; if y = 5, X = 4; if y = Si X = 6 ; etc. It is evident that an unHmited num- ber of values may be given to y and x that will satisfy the equation. Hence, an ecj nation containing two unknown numbers admits of as many solutions as we please, and is indeterminate. Since the values of the unknown numbers are dependent upon each other, they may be confined to a particular limit ; as, for exam- ple, suppose the variables to be restricted to positive or negative inte- gers, we may thus limit the number of solutions. Example 1. Solve 19x-i-by= 119, in positive integers. Solution. Transpose 19 a:, 5i/=119— 19 2:. Therefore, y = 23-3x + 4l-^] (1) l-x Since the value of y is to be integral, then - must be integral, although fractional in form; and so also is any multiple of it. Let — r— = n, an integer. Therefore, x=l-5n (2) Substitute in (1), y = 20+19 n (3) We must take only such integral values for n as will give positive integral values for x and y. (2) shows that n may be 0, or have any negative integral value, but cannot have a positive integral value. 356 ELEMENTS OF ALGEBRA. (3) shows that n may be and — 1, but cannot have a negative integral value greater than 1 . Therefore, n may be and — 1. ^^"^^'^=2i}'^^'%=i}' Query. Can n be — 2 or + 1 ? Why ? Example 2. Solve 7 x — l^y = 19, in positive integers. Process. Transpose and solve for a;, x = 2-hy + bl — ;=— J (I) Let —j — = n, an integer. Therefore, y = 7n-l (2) Substitute in (1), x=l2 7i-\-l (3) Evidently x and y will both be positive integers if n have any positive integral value. Hence, x = 13, 25, 37, 49, ... . y= 6, 13,20, 27, .... Notes : 1. Having obtained a few of the possible values of x and y, the law will become evident. 2. It will be seen from the above solutions that when only positive integral values are required, the number of solutions will be limited or unlimited ac- cording as the sign connecting the terms is positive or negative. Example 3. Solve I90x — 23y = 708, in least positive integers. /3 - x\ Process. Solve for y, y = Sx~30 — 6\—^j (1) S — x Let ^ = n, an mteger. Therefore, x = 3-2Sn (2) Substitute in (1), y = -6-190 n (3) Evidently x and y will both be least positive integers if n be — 1 . Therefore, n = — ], x = 26, and y = 184. Note 3. If the coefficient of the unknown number in the numerator of the fraction is not 1, it will be necessary to make several transformations. Example 4. Solve 21a:+ 17 y = 2000, in positive integers. » ._ — ", ail xiiiegei. 3-n (2) 3-»j — 7 — = w, an integer. n = 3 - 4 m. x= 17m-10 (3) 3/= 130- 21 m (4) INDETERMINATE EQUATIONS. 357 11 -4x Solution. Solve for y, y = 117 - x + — r= — (1) w U-4x Transpose, y + a: — 117 = — yj Since x and y are to be integral, y + x — 111 will be integral ; 11 — 4 a: .,, , , hence, — r^ — will be integral. ll-4a: Let Therefore, >.x 3-n , . Now —z — mwtt be integral. Let Therefore, Substitnte in (2), Substitute in (1), (3) shows that m may have any positive integral value, but can- not be 0, or have any negative integral value. (4) shows that m may have any integral value from to 6, or any negative integral value, but cannot have a positive integral value greater than 6. Therefore, m may be 1, 2, 3, 4, 5, Q, giving the following pairs of values : x= 7, 24, 41, 58, 75, 92. y = 109, 88, 67, 46, 25, 4. Hence, To Solve a Simple Indeterminate Equation, Involving Two Unknown Numbers, for Integral Values. Find tlie value of one of the unknown numbers. Place the fractional part of this value efjual to Uj an integer, and solve the resulting equation for the other unknown number. Substitute this result in the value first obtained. Solve the two simple equations thus fonuerl, by inspection, for inte- gral values of n. Notes: 4. It is better, in solving the original equation, to solve for the unknown number which has the least coefficient. 5. A little insrenuity in arranging the terms will often obviate the necessity of a second transformation. 358 ELEMENTS OF ALGEBRA. 148. There can be no integral values of x and y in an equation of the form ax ±h y = c, ii a aiid b have a com- m6n factor not common also to c. For, suppose d to, be any factor of a and also of b, but not of c, sucli that a — md and h = n d. c Then mdx ±.ndy = c, or mx ztny = -j. Since m and n are integers, if 'x and y be also integers, mx ±. ny is an integer. But ^ is a fraction. Hence, no integral values of X and y can be found. Notes : 1. If a, b, and c have a common factor, it should he removed by division, then proceed as in Art. 147. 2. The solution of any indeterminate equation of the form ax — by =. ±c, in which a and h are prime to each other, is always possible, and admits of an unlimited number of integral solutions (Ex. 2, Art. 147). If the equation be of the form ax-\- hy = c, the number of results Avill always be limited ; and, in some cases, the solution is impossible (Ex. 1, Art. 147). Exercise 130. Solve in positive integers : 1. 2x-\-3y = 2o; Ux = 5y-7; 3x = 8y-16, 2. 5r?:+ ll2/ = 254; 9 a: + 13 2/ = 2000. 3. 15x-l7y=l; 13x-9y=:l; 9x-Uy=:10. Solve in least positive integers : 4 3^ + 7y = 39; 3x+4y=:39; 7x+lDy = 22^. 5. 27a;-192/ = 43; 2^ + 7?/ =125; 555y-22a: = 73. 6. 19 ^-5?/ =119; I7x = 4:9y-S. Are integral solutions possible for the following? Why? 7. 3^ + 21 7/ = 1000; 7 a; +14?/ = 71. 8. 323 a?- 527 y= 1000; 166 a: - 192 y = 91. INDETERMINATE EQUATIONS. 369 9. Solve 7 a- 4- 15 y = 145, in positive integers, so that X may be a multiple ot y. ^ 146 , 145 n Suggestion. Let x-ny, then y = ^ ^^ , and x = ^^^^^ . 10. Solve 'S9 X — 6 y = 12, in positive integers, so that y may be a multiple of z, 11. Solve 20 a: — 31 2/ = 7, so that x and y may be positive, and their sum an integer. Suggestion. Put x + y = n. 149. A problem is indeterminate when it involves less conditions than there are unknown numbers. Exercise 131. 1. Find a number which being divided by 3, 4, and 5, gives the remainders 2, 3, and 4, respectively. Solution. Let x represent the number and y the sum of the quotients, then, x-2 a--3 x-4 3 +^ + — = »• f'S-\-y\ Simplify and solve for ar, a- = y + 2 4- 13 I — j;^ I (1) 3-l-w \ •*' / Let -^=- = n, an integer. Therefore, y = 47 n — 3. Substitute in (1). a: = 60 n - 1. Hence, n may be 1, 2, 3, 4, etc. Therefore, x = 59, 119, 179, 239, etc. .7 = 44, 91, 138, 185, etc. 2. Find the least number which being divided by 2, 3, 4, 5, and G, gives remaindei-s 1, 2, 3, 4, and 5, respectively. 3. Find two numbers which, multiplied respectively by 14 and 18, have for the sum of their products 200. 360 ELEMENTS OF ALGEBRA. 4. Divide 142 into two parts, one of which is divisible by 9, and the other by 14. 5. There are two unequal rods, one 5 feet long and the other 7. How many of each can be taken to make up a length of 123 feet ? 6. Find two fractions having 5 and 7 for denominators, and whose sum is ||. 7. Find the least number that when divided by 9 and 17 will give remainders 5 and 12, respectively. N- 5 Suggestion. Let N represent the number, — - — = x, and N~ 12 — p^ — = y. .'. 9x=l7y-{-1. 8. A farmer bought sheep, pigs, and hens. The whole number bought is 125, and the whole price, $225. The sheep cost $5, the pigs $2.50, and the hens 25 cents. How many of each did he buy ? Solution. Let X = the number of sheep, y = the number of pigs, and z = the number of hens. Then, x + y + z= 126 (1) and 5x+2. ■5y + .25 z = 225 (2) From (1) and (2), ?/=86-2x-^ (3) Let x-1 Q - n, an integer. Therefore, x = 9n-\-l. Substitute in (3), y = M-l9n. Substitute in (1), 2;= 40+ 10 n. Therefore, n may be 0, 1, 2, 3, and 4, giving the following values : x= I, 10, 19,28,37. y = 84, 65, 46, 27, 8. z = 40, 50, 60, 70, 80. PROBLEMS. 361 Qaeries. How many solutions ? In how many diflFerent ways may the stock l)e bought? How solve by means of only two un- known numbers? 9. How can one pay a sum of $ 1.50 with 3 and 5 cent pieces ? In how many ways can the sum be paid ? 10. Can a grocer put up the worth of S3.50 in 11 and 7 cent sugar ? In how many ways can he do it in even and odd pounds, respectively ? Find the greatest and least number of pounds of the 7- cent sugar he can use. 11. Is it possible to pay £50 by means of guineas and three-shilling pieces only ? 12. A owes B $5.15. A has only 50-cent pieces and B only 3-cent pieces. How may they settle the account ? 13. A farmer bought horses at S 60 a head and sheep at S8, and found that he bad invested S4 more in sheep than horses. How many of each kind did be buy ? 14. A farmer invested $1000 in 75 head of cattle, worth $25, $15, and $10 per head. Find the number of each kind, and the number of ways in wliich he could buy them. 15. A grocer had an order for 75 pounds of t€a at 55 cents a pound, but having none at that price he mixed some at 30 cents, some at 45 cents, and some at 80 cents. How much of each kind did he use, and in how many ways can he mix it? 16. How many pounds of 20, 35, and 40 cent coffee must a grocer take to make a mixture of 150 pounds worth 30 cents a pound ? In how many ways can the mixture be made ? 362 ELEMENTS OF ALGEBRA. 17. How many gallons of S 1.50, S1.90, and $1.20 wine must a vintner take to make a mixture of 40 gallons worth $1.60 per gallon ? How many ways may the mixture be made ? Can an odd number of gallons of each kind be taken ? An even number ? 18. In how many ways can £1 be paid in half-crowns, shillings, and sixpence, the number of coins in each pay- ment being 18 ? 19. A hardware merchant paid $180 for 20 stoves. There were three sizes: one $19 each, another $7, the other $6. How many of each size did he buy? 20. A person having a basket of oranges, containing between 50 and 72, takes them out 4 at a time, and finds 1 over; he then takes them out 3 at a time, and finds none over. How many had he ? A^-l N Suggestion. Let N represent the number, — j— = x, and -« = t/. l+x ^ I + x .'. y = x+ ~^' Pwt — ^ = n. Then n must be 5 or 6. 21. A poultry dealer has a basket containing between 200 and 300 eggs, he finds that when he sells them 13 at a time there are 9 over, but when he sells them 17 at a time there are 14 over. Find the number of egfjs. 22. Two countrymen together have 100 eggs. If the first counts his by eights and the second his by tens, there is a surplus of 7 in each case. How many eggs has each ? 23. A surveyor has three ranging poles of lengths 7 feet, 10 feet, and 12 feet. How may he take 40 of tliem to measure 113 yards? In how many ways may the mea- surement be made ? INEQUALITIES. 363 CHAPTER XXV. INEQUALITIES. 150. Since a positive number is greater than any negative num- l)er, the statement that a is algebraically greater than 6, or that a—h is positive, is expressed by a > 6 ; that a is algebraically less than 6, or that a — 6 is negative, is expressed by a < 6. Hence, An Inequality is a statement that on^ expression is greater or less than another; as, 1 — ar > = — : m — n < x. The expression at the left of the sign is calle<l the first member, and the expression at the right, the second member of the in- equality. The fonn a> h> c, means that 6 is less than a but greater than c. Notes : 1. Inequalities are said to subsist in the same sense wlien the lirst member is the greater in each, or the first member is the less in each ; as, 3 > 2, 7 > 5, and .5 > 3 ; a<h, c<d, and m<n. 2. Two inequalities are said to subsist in a contrary sense when the first member is the greater in one, and the less in the other ; as, 5 > 3 and o < 6 ; m < 5 and h > n. 3. An inequality is said to be solved when the limit to the value of the unknown numl)er is found. 151. Subtract a + & from each member of a > 6, then, a - (a + b) > b- (a -\- 6). Simplify, . — 6 > — a, or, ~ a < — b Hence, I. If each member of an inequality has its sign changed, the sign of inequality will be reversed. 364 ELEMENTS OF ALGEBRA. Multiply each member of - 5 < 5 by - -2, then, 10 > - 10. Multiply each member of a > 6 by - -m, then, — am < — 6m. Divide each member of - 6 < 4 by - -2, then, 3 > - 2. Divide each member of a > ft by - -m, then. a b m m' He II. If each member of an inequality be multiplied or divided by the same negative number^ the inequality tvill be reversed. Suppose a> b, c > d, m'> n, — By definition, a — b, c-d, m-n, are positive. Add, (^a -b) + {c - d) + (m - n) + .... is positive. or, (a + c + m -I .) - (b + d -\- n -\- ... .) is positive. Therefore (by definition), a + c -^ m + — > b + d + n + .,», Thus, 7 > 3 5> 2 4> 1 Add, 16 > 6, or divide by 2, 8 > 3. Hence, III. If the corresponding members of several inequalities be added, the sum of the greater members will exceed the sum of the lesser members. Suppose a > b and m > n, then a - b and m — n are positive. But, (a — b) — (m — n), or (a — m) — (6 — n) may be either positive, negative, or 0. Therefore, a — m > b — n, a — m < b — n, or a — m = b — n. Thus, 5 > 3 3> 2 Subtract, 2 > 1 7>4 5> 1 Subtract, 2 < 3 8> 7 6>5 Subtract, 2 = 2. Hence, IV. If the members of one inequality be subtracted from the corres- ponding members of another, tht resulting inequality will not always subsist in the same sense. INEQUALITIES. 865 1 2 X 3 z 64 Example 1. Solve 3^ x ^ — > -^^ + ^^ for the limits of x. Solution. Free from fractions and simplify, 112a;-6>45x+ 128. Subtract 45 x - 6, 67 a; > 134. Divide by 67, x > 2. Therefore, x is greater than 2. Example. 2. Solve the following : \bx-Zy>Zx-^b (1) i3a: + t/=r22 (2) Solution. Subtract 3 x from (1), 2 x - 3 y > 5 (3) Multiply (2) by 3, 9x + 3 y = 66 (4) Add (3) and (4), 11 x > 71. Divide by 11, x > Q^^. 22 -w From (4), x = ^ • Substitute in (3) and simplify, —y>-\\. Therefore, 2/ < 2^^ (see I) Example 3. Solve the inequalities : {fix — mn^n^ — mx (1) \ mx ~ nx + mn < in^ (2) Process. Simplify (1) and solve, x > n. Simplify (2) and solve, x <rr. Hence, x is greater than n and less than m. Note. The principles applied to the solutions of equations may be applied to inequalities, except that if each member of an inequality haa iti sign changed, the lign of inequality will be reversed. Exercise 132. Find the limit of x in the following : 1. 4x-3 >fa;-f; f — | ^ < 9 - 3 a:. ^o o o 1 x^ — a a — X 2x a 366 ELEMENTS OF ALGEBRA. o ax — 2b a X — a ax 2 ^- Zh 2h~ ^ 1" ~ 3 * 4. If 2^2 + 4 a; > 12, show that x>2. 5. If 7 a;2 - 3 a; < 160, show that x < o. 6. If 4 ic + 12 — ic^ > 0, show that x is included be- tween 6 and — 2. 7. If 9 a; < 20 it^ + 1, show that x > \ oi < \. 8. If 15 — ic — 2 ic^ > 0, show that x lies between | and — 3. Q I ^ ;^ > 30 — 4 ic. ^Q ( 1 J ^ < I ^ + 3|-. < 3 :?: + 49. * ( 6 oj > 24 - 2 ^. Find an integral value of x in the following \ IQx ( |(^ + 2) + ^ < I (^ - 4) + 9. ' \l(x-\-2) + \x>l{x+l)^^. Find the limits of x and y in the following : .o i3^4-5^>121. |7.^; + 5^>19. * (4a;+ 72/ =168. / a; - 3/ = 1. (« + &) ^ — (a — ?)) 2/ > 4 a &. ^ I (tt - &) ^ + (a 4- h)y = 2(a + h) {a - 6). 15. A certain number plus 5, is greater than one third the number plus 55 ; while its half plus 2, is less than 41. Find the number. INEQUALITIES. 367 16. Find the price of oranges per dozen, when three times the price of one orange, decreased by tiiree cents, is more than twice its price increased by one cent ; and eight times the price of one orange, decreased by twenty cents; is less than three times its price increased by ten cents. 152. Since the stjuare of a negative number is positive, if a and b represent any two numbers, (a — 6)^ must be positive, whatever the values of a, and b. Therefore, since every positive number is greater than zero, (a - 6)2 > 0. Expand, a^ - ^ah + h'^> 0. Add 2 a 6 to each member, a* + 6^ > 2 a 6. Hence, The sum of the squares of two unequal numbers is greater than tvnce their product. Vote. The above is a fundamental principle in inequalities. Example 1. Show that a"^ -\- 1)^ ■\- c^ > ab + a c + h c, a and h positive. Prool Since a, b, and c are any unequal numbers, a2+62>2a6 (1) a«+c«>2ac (2) 62 + c«>26c (3) Add the corresponding members of (1), (2), and (3), 2aa + 2 62-l-2c2>2a6+2ac + 2 6c. Divide by 2, a« + 6« + c^ > a 6 + a c + 6c. Query. How if a = 6 = c ? Example 2. Show that a« + 6« > a* 6 -}- a 6*. Proof. We shall have, a* + }fi > a'^b + a 6*. Factor, (a + 6) (a« - a 6 + 6^) > a 6 (a + 6). Divide by a + 6, a'^ — ah -\- b"^ > ab. Add a 6, a« + 6^ > 2 a 6. Therefore, a« + 6» > a^ 6 + a 6*. 368 ELEMENTS OF ALGEBRA. Example 3. Which is the greater, V/ 1- 1/ — r- or /y/a i Proof. We shall have, Square each member, a2 62 2 Y a bmn -\ r- > or < a 6 + 2 \/a bmn + m n. it9f tV CI O , a^h^ m^n^ Subtract 2 \/ahmn^ 1- — r- > or < ao + ww. ^ m n a Free from fractions and factor, {ah + m n) {a^ h^ — abmn -\- m^ n^) > or < abmn(ah + mn). Divide by ab -\- mn, a^h^ — abmn + m^n^ > or < ab mn. Add a b 7« n, a^ b^ + m^ n^ > or < 2 a 6 m n. But, a^b^-i m'^n^':> 2abmn. Therefore, \ 1^ + \ -afT > ^""^ -^ V^""- Exercise 133. Show that, the letters being unequal, positive, and integral : h^ a^ a b 2. a 6-2 + a-% > a-^+ b-^ ; (ni^-]- n^) (mH n^) > (m^+ n^f. 3. xy-\-xz-\-yz < {x + y^z)^-\-{x+z—yf'-{-{y + z~x)\ Which is the greater : . „ . „ , a + b 2 ab m n 11 4. 71^4-1 or ?i2+ ^ ; —^ or —-7 ; -5 s or 5. 1±J or ^^; 3(1 + a2 + a^) or (1 + c^ + a'f, ga^ y x^ — y^ ^ INEQUALITIES. 369 1/9 a/^ V3 v5 Queries. How in 4 and 6, if a = 6 ? In 4, if n = 1 ? 7. If a^ + ^>2 + c2 = 1, and a?» + y» + «2 = 1, show that aa; + &y + C2 < 1. Query. How ifa = 6 = c = a: = y = z? If a > 6, show that : 8. a - 6 > iVa - V6)^ a^ + 7 aH > (rr + &)» 9. a-6* > a*6« ; a^ + 13 a ^2 > 5 a26 + U 63. Miscellaneous Exercise 134. Example 1. Solve the inequahties : {: V2(xy-f-y2) + 4<;y(2i/-l)(y + a:) (1) 2x + 5y>8 (2) Solution. Square each member of (I) and simplify, 2xy-\-2y^-\- 4 <2y^-^2xy -y-x. Subtract 2ari/ + 2 y«, 4 <-y-x (3) Multiply each member of (3) by 2, 8<-2y-2a: (4) Add the corresponding members of (2) and (4), 3y > 16. .-. y >5f Multiply each member of (3) by 5, 20<-5y-5a: (5) Add (2) and (5), - 3a: > 28, or 3a: < -28. .-. x < -9f Example 2. Simplify (y-\-x<m — n) (m* + m n 4- 71^ > y — x). Solution. We are to multiply the corresponding members together, (y + x) (y — x) = if — a:*, (m — n) (m* ■\- mn -\- n'-*) = m* — n*. Therefore, (y +x < m-n){mHmn-\- n^ >y-x) = y^~x^ < m*- n». 24 370 ELEMENTS OF ALGEBRA. Example 3. Which is the greater, x^ + y^ or x^y -\- y*xl Proof. We shall have, x^ + y^> or <: x* y -{- y* x. Subtract x^y -\- y^x, x^ — x^ y + y^ — y* x > or < 0. Factor, (x^ - y^) (x-y)> or < 0. Now, whether a; > or < .y, the two factors, x^ — y^ and x — y, will have the same sign. Hence, since (x'^—y^) i^~y) is always positive, {x^ - y^) {x-y)> 0. Therefore, a;^ + 2/^ > x^ y -\- y^x. Example 4. Which is the greater, m* — n* or 4 n^ (^ _ ,^) when m > n ? Proof. We shall have, m^ — n* > or < Am\m—n). Divide by w — n, m^ + wi*n + m w^ + n^ > or < 4 m^. Subtract wi^ + m^ n and factor the resulting inequality, n^ (m + n) > or < m\^m—n). But, m > w (1) Square (1), m^ > n^ (2) Multiply (1) by 2, 2 m > 2 n. Add m — n, 3m — ??>m + n (3) Multiply the corresponding members of (2) and (3), m^ (3 m — n) > n' (m -{- n). Therefore, 4m^ (m — n) > w* — n*. 6. Find the sura of x^ + y > 1 — a, y^ — 2 a > 5 + 4, ^ X + y < 2 a + 1, and y^ - S x^ < 5 - a. 6. From a^ + 2 a a;^ < 5 take a (a + x^) y n^ —1. 7. From a2 < 3 - .7.2 subtract 2 ^2 > 5. Multiply : 8. {a + hf > {x-yf by -3; Z-f < 5-^^ by x^ + y\ 9. Divide a^-l^ > a^+ h^ by a2 + ^2 10. Divide 11 a2 + 88& > 121^2 ^y - H. INEQUALITIES. 371 Perform the indicated operations and simplify : 11. (w-l< 5)(m + l<10); (a < n + h){n-b > c). 12. (_ 2 > - 3)3 ; (5 > 2) -f- (3 < 4) ; V25 > 9. 13. [- 243 > - 1024]i ; (71 + 1)^ > n^ - n^ + 4: n. 14. m3 - 7i3 > (m - n) {m^ + n^) ; 4^- 64 < 8. 15. (m2 - n^ < u.^)'T-{ix> m + n); [- ?t > i/f. Solve : 16. {X''2f > 0^+ 6 x-25',V(x-l)^ + 'S 2^ + 6 >2xK 17. a; - 2 > V ^-^=^ ; V3 - 4 Vi > VI6 2: - 5. ^g |3y+2a;>3. ^^ | a: + f > Vo^ - 3 a; + y. ( 4 > 4 7/ + ic. ■ ( 5 > a: — y. j42/-a^>?/ + 4. J3:r-l>a: + 3y. ^"- (3a;-6y> l-4i/. "^ ( 27/-3rr2 = 3a;-3.x2. 22. 38a:-7-15a:2<0; 6 ar^ 4- 7 ;r + 2 < 0. 23. 17 a;- 6r»-5<0; 6 a: + 11 - a:^ < 2 a; - 10. Find integral values of x in the following : (3ix-.5x>5. {x + 7 ^^- l2.5a: + ia;<8. ^^' (2a; + a; + 7 < 15. 10 > 20. 26 U^-i^<3. 27 |2a:-5>31. I 7a;-15>4a;+30. ^' (3a;-20<2ar. 28. ar2 4- 2a;-15<0; a:^ ^ lOa: 4. 63 < 0. 372 ELEMENTS OF ALGEBRA. Show that: 29. Vl9+V3> VIO + VT; V5 + Vn > V3+3V2c li a >h, show that : 30. Va^ - 62 + Va^ - (a - hf > a. 31. a^-h^ <3a^{a- h) and >Sb^{a- b). 32. a-h> -j-^ and < -^3- . If x^ — a^ + y^, if' — (p- -^ d'^^ show that: 33. xy'^ac + bdoTad + bc. Show that: 34:. {a b + X I/) (a X + b y) ^ 4: a b xy, 35. (ti + 6) (ft + c) (6 + c) > 8 a & c. 36. Show that the sum of any fraction and its reciprocal is greater than 2. ?7. In how many ways may a street 20 yards long and 15 wide be paved with two kinds of stones ; one kind being 3f feet long and 3 wide, the other 4| feet long and 4 wide ? 38. A and B set out at the same time to meet each other; on meeting it appeared that A had travelled a miles more than B, and that A could have gone B's distance in n hours, and B could have gone A's distance in m hours. Find the distance between the two places. Solve when ft = 18, ^ = 378, and m = 672. SERIES. 373 CHAPTER XXVI. SERIES. 153. A Series is an expression in which the successive terms are formed according to some fixed law ; As, 1, 2, 4, 8, , in which each term is double the preceding term ; a, a + c/, a + 2d, a + 3d, , in which each term exceeds the preceding term by d. ARITHMETICAL PROGRESSION. 154. The expressions 1, 6, 9, 13, 17, , and 16, 10, 6, 0, — 5, —10, ...., are called arithmetical progressions or series. The first is an increasing series, and the second a decreasing series. The general form for such a series is, a, a + d, a-\-^d, a + 3<i, a + 4ci, a + Hd, a + 6cl, .... in which a is the first term and d the common difference ; the series will be increasing or decreasing according as d is positive or negative. Hence, An Arithmetical Progression is a series in which the adja- cent terms increase or decrease by a common difference. In every arithmetical series the following elements occur, any three of which being given, the other two may be found : The first term, or a. The last term, or /. The common difference, or d. The number of terms, or n. The sum of the terms, or s. 374 ELEMENTS OF ALGEBRA. By an examination of the general form it is seen that the coefficient of d is always 1 less than the number of the term. Thus, the 2d term is « + rf, or a + (2 — 1) cf, 3d term is a + 2 </, or a + (3 — 1) rf, 4th term is a f- 3 </, or a + (4 — 1) 6?, 12th term is a + 11 rf, or a + (12 — 1) d, and so on. In the nth, or last term, the coefficient of c? is n — 1. Hence, To Find the Last Term of an Arithmetical Series, when the first term, the common difference, and the number of terms are given. I = a+ {n-l)d (!) Note. The common difference may always be found by subtracting any term of the series from that which immediately follows it. Example 1. Find the 18th term of the series |, f, |, etc. Process. Here, n = 18, a = |, and d — ^ — l = \. Substitute these values in (i), / = | + (18 - 1) J = 4. Example 2. Find the 30th term of the series x -\- y, x, x — y, etc. Process. Here, n = 30, </ = x — {x+y) = —y, and a = x + y. Substitute these values in (i), I = x + y + (30 — l)(—y) = x — 28y. Exercise 135. Find: 1. The 15th term of 7, 3, - 1, .... 2. The 27th and 41st terms of 5, 11, 17, .... 3. The 20th and 13th terms of - 3, - 2, - 1, .... 4. The 37th and 89th terms of- 2.8, 0, 2.8, .... 5. The 40th term of 2 a — h, 4 a - 3h, 6 a — 5h, .. 6. The 15th and 8th terms of J, 1 . | ARITHMETICAL PROGRESSION. 375 7. The first term is J, the 102d is 18. Find the com- mon difference. 8. The 21st term is 53, and the common difference is — 2J. Find the first term. 9. The first term is 5^, and the common difference is 3 J. What term will be 42 ? 10. The first term is ^, the common difference is ||, and the last term is 17^. Find the number of terms. 11. The 54th and 4th terms are - 125 and 0. Find the 42d term. 12. Find three terms whose common difference is J, such that the product of the second and third exceeds that of the first and second by. 1 J. 155. Taking the elements as given in Art. 154 : s = a + (a + rf)+(« + 2</) + (a + 3rf)+(a + 4f/) + ••.. ^ or 5 = / + (;-r/)-f (/-2c?H(/-3£/) + (/-4cO + .-.. a- Add, 2s = {a + [)-\-{a + l)-\-{a-\-l) + (a + l) + (a + l) -f .... to n terms, or 25 = n(a + /) (1) Substitute the value of I from (i) (Art 164) in (I), a » = n [2 a + (n — 1) d]. Hence (solve for s), To Find the Sum of all the Terms of an Arithmetical Seriei » = ^[aa-f (n-l)<f] (iH) Example 1. Find the sum of an arithmetical series of 17 terms, the first term being 5^, and the last term 25 i. Process. Here, n = 17, a = 5^, and / = 2.''>|. Substitute these values in (ii), « = y. (5^ + 25^) = 263j. 376 ELEMENTS OF ALGEBRA. Example 2. Find the sum of the series 3|, 1,-1^, , to 19 terms. Process. Here, n = 19, a = 3^, and c? = 1 — 3^ = — 2^. Substitute these values in (iii), s= ^-[2 X 3^ + (19 - 1)(- 2^)] = - 361. 12 3 Example 3. Find the sum of m , 3 m , 5 m , . . . . , to m' m m' ' m terms. Process. Here', n = m, a = m — — , and d =3m —~ ~ m — -) = 2m — - • m Substitute in (iii), 2 m^ — m — 1 =fK'"-9+^'" -'>("" -9] Example 4. The first term of a series is 3 m, the last — 35 m, and the sum — 320 7W. Find the number of terms and the common di Herence. Process. Here, s =i — 320 m, a = 3 m, and / = — 35 m. Substitute in (ii), - 320 m = ■x(S m - 35 m) = - 16 mn. .-. n = 20. Substitute in (iii), - 320 m = %0- [6 m + 19 rf] = 60 m + 190 c?. .-. d = -2m. Example 5. How many terms of the series — C|, — 6f , —6, , must be taken to make - 52| ? Process. Here, s= — 52|, a = — 6|, and c/ = f . n Substitute in (iii), - 52| = ^ C" ¥ + (n - 1) X f ]. Simplify and solve for n , n = 1 1 or 24. Query. Do both of these values satisfy the conditions ? In explanation write out 24 terms of the series and observe that the last 13 terms destroy each other. ARITHMETICAL PROGRESSION. 377 Exercise 136. Find the sum of : 1. 5, 9, 13, ..... to 19 terms. 2. 10 J, 9, 7 J, ...., to 94 terms. 3. 3 a, a, — a, . . . . , to a terms. 4. 3 J, 2 J, 1|, ...., to n terms. ^w— Im — 2 m — 3 5. , , , ...., to m terms. m mm . 2ag~l ^ 3 6 g^ - 5 6. , 4a , , ..... to n terms. a a a ^ 4a + & 5a + 2& 7. a, — X — , ^ , ...., to 19 terms. 8. The first term is 3^, and the sum of 14 terms is 84J. Find the last term. 9. The sum of 40 terms is 0, and the common difference is — ^. Find the first term. 10. Find the number of terms and common difference: (1) when the sum is 24, the first term 9, and tlie last —6; (2) the sum 49 a, the first term a, and the last 13 a. 11. The sum of 12 terms is 150, and the first is 5J. Determine the series. 12. Show that the sum of the first n odd numbers is r?, 13. Find the sum of all the odd numbers between 100 and 200. 378 ELEMENTS OF ALGEBRA. 14. The sum of five terms is 15, and the difference of the squares of the extremes is 96. Find the terms. 15. Find the sum of -=i, tj , 7=, ...., l'\- Vx ^-^ 1- Vx to n terms. 156. a is called the arithmetical mean between a — d and a + d. Hence, An Arithmetical Mean is the middle term of three num- bers in arithmetical series. If a and 6 represent two numbers, and A their arithmetical mean, the common difference is A — a, or b — A. Therefore, A — a — h — A. Hence (solve for J.), To Find the Arithmetical Mean Between two Terms. A = ^ M If a and I represent any two numbers, and m the number of means between them, the whole number of terms is m + 2, or wi + 2 = n. Substitute this value for n in (i) (Art. 154), I = a + (m + I) d. Hence (solve for d), To Insert any Number of Arithmetical Means Between two '^«"-- ^ . 1- (V) This finds d, and the m required means are, a-\- d, a + 2d, a + ^d, a + 4Ld, ...., a-\-md. Example 1. Find the arithmetical mean between: (1) 27 and — 5 ; (2) rri^ -\- mn — n^ and m^ — m n -\- n^. Process. (1) Here, a = 27, 6 = — 5. 27-5 Substitute in (iv), A = — -x — = 11. (2) Here, a — m^ + mn- n^, b =m^ — mn + rfi. ^ , . . ,. ^ , m^ + mn -- n^ + m^ — mn + n^ Substitute in (iv), A = 2 — =; m\ GEOMETlilCAL TUOGRESSION. 379 Example 2. Insert five arithmetical means between 12 and 20. Process. Here, a = 12, / = 20, and m = 5. 20- 12 Substitute in (v), d = ^ ^ = H- The series is 12, 13^, 14|, 16, 17^, 18f, 20. Exercise 137. Insert : 1. 14 arithmetical means between — 7-J and — 2^. 2. 16 arithmetical means between 7.2 and — 6.4 3. 10 arithmetical means between 5 m— 6 n and 5w— 6 m. 4. 4 arithmetical means between — 1 and — 7. 5. X arithmetical means between a^ and 1. « ^. , , . , . , , m— n , m-hn 6. Find the arithmetical mean between — — - and . m+n m — n 7. The arithmetical mean between two numbers is — 9, and the mean between four times the first and twelve times the second is — 66. Find the numbers. GEOMETRICAL PROGRESSION. 157. The expressions 3, 9, 27, 81, ...., and 1, -i, i, -^, ...., are called geometrical progressions or series. The general form for such a series is, a, ar, ar^, ar*, ar*, ai*, an*, ar'', ...., In which a is the first term, and r a constant factor or ratio. Hence, A Geometrical Progression is a series in which the adja- cent terms increase or decrease by a constant factor. The Common Ratio is the fiictor by which each term is multiplied to form the next one. 380 ELEMEJ^TS OF ALGEBRA. In every geometrical series the following elements occur; any three of which being given, the other two may be found. The first term, or a. The last term, or I. The common ratio, or r. The number of terms, or n. The sum of the terms, or s. By an examination of the general form it is seen that the expo- nent of r is always 1 less than the number of the term. Thus, the 2d term is a r, 3d term is a r^, 4th term is a r*, 12th term is ar", and so on. In the nth, or last term, the exponent of r is w — 1. Hence, To Find the Last Term of a Geometrical Series, when the first term^ the comtnon ratio, and the number of terms are given. I = ar""-^ (i) Notes: 1. The common ratio is found by dividing any term by that which immediately precedes it. 2. A geometrical series is said to be increasing or decreasing, according as the common ratio is greater than 1, or less than 1. 3. An arithmetical series is formed by repeated addition or subtraction; a geometrical series by repeated multiplication. Example 1. Find the 8th term of the series .008, .04, .2, etc. Process. Here, a = .008, n = 8, and r = .04 ^ .008 = 5. Substitute in (i), / = .008 X 58-i = 625. Example 2. Find the 10th term of — , x, y, — , Process. Here, a = — , n = 10, and r = x -. — = - • y\ ' y X Substitute in (i), / = - f|j = ar-'yS GEOMETRICAL PROGRESSION. 381 Exercise 138. Find: 1. The 5th and 8th terms of 3, 6, 12, .... 2. The 10th and 16th terms of 256, 128, 64, .... 3. The 8th and 12th terms of 81, - 27, 9, .... 4 The 14th and 7th terms of gJ^, ^^, 3^, .... .- rr.1 « 1 ^ X mx m^x 5. The 6th term of -, — 5-, — 3- y / f 6. The mth term of x, x^^ 7^, .... 7. The 3d and 6th terms are f^ and — |. Find the series and the 12th term. 8. The 5th and 9th terms are f J and §. Find the series. 9. If from a line a inches in length, one third be cut off, then one third of the remainder, and so on; what part of it will remain when this has been done 5 times ? When / times. 168. Taking the elements as given in Art. 157, 8z=a + ar-\-ar^ + ar*-\- -\-ai*-^-\-ar^'^ (1) Multiply (1) by r, sr = ar-i-ar^-i-aH»-| -f- a r"-'^ -I- g r»-^ -t- g r (2) Subtract (1) from (2), sr—8 = ar^ — a (3) Substitute the value of or* from (i) (Art. 167) in (3), and factor the result, « (r - 1) = r / - a. Hence (solve for <), To Find the Sum of all the Terms of a Geometrical Series 8 = ^^null (ii) r - 1 382 ELEMENTS OF ALGEBKA. Example 1. Find the 6th term and the sum of — J, ^, - f , . Process. Here, a = —^, 7i = 6, and r = — |. Substitute in (i) (Art. 157), ^ = - |^ x (- |)^ = |^. Substitute in (iii), 5 = " '^_^s _ ^ ^ = W- EXxiMPLE 2. Find the least term and the sum of 3, — 9, 27, to 7 terms. Process. Here, a = 3, n — 1, and r — — 3. Substitute in (i) (x\rt. 157), Z = 3 (- 3)« = 2187. 3^:^y— =1641. Substitute in (ii), ^= -3 Exercise 139. Find the sum of: 1. 3, -1, J,.... , to 6 terms. 2. -lh-i>' ,..., to 6 terms. 3. 1, -J, A. •• . ., to 8 terms. 1 3 *• V3' ' V3' ...., to 8 terms. 5. 1, 3, 32, ..... to m terms. 6. 2, -4, 8, ... . , to 2 m terms. 7. The 7th and 4th terms are 625 and — 5. Find the 1st term, and the sum of the 4th to the 7th terms inclusive. 8. The sum of the first 10 terms is equal to 33 times the sum of the first 5 terms. Find the common ratio. 9. The sum of three numbers in geometrical progression is 216, and the first term is 5. Find the common ratio and the numbers. GEOMETRICAL PliOGRESSION. 383 159. A Geometrical Mean is the middle term of three numbers in geometrical series. If a and b represent two numbers, and G their geometrical mean, G b the common ratio is — , or ^. Therefore, G b — = jy. Hence (solve for G), To Find the Geometrical Mean Between two Terms O = \/ab ' (iv) If a and b represent any two numbers, and m the number of means between them, the whole number of terms is m + 2, or m -f- 2 = n. Substitute this value for n in (i) (Art. 167), / = a r"* + 1. Hence (solve for r), To Insert any Number of Geometrical Means Between two Terms. , , _ This finds r, and the m required means are, ar, ar^, ar^, ar^, ar^ , ar»*. Example 1. Find the geometrical mean between : (1) — -= and 3 V3 7^; (2) 3x»y and I2xfz. "^^ 1 3 Process. (1) Here, a = -—p, and b = —-z • Substitute in (iv), G = v/ — p ^ ^=^ 3^ V3 V3 (2) Here, a = 3x*y and b = Uxy'z. Substitute in (iv), G = ^33*y X I2xy*z = 6x^y^ ^z. Example 2. Insert six geometrical means between 14 and - ^y. Process. Here, a = 14, / = - /y, and m = 6. Substitute in (v), r = ^—^\^ = — i- Hence, the series is 14, -7, f - }, |, -^^h- ^. 384 ELEMENTS OF ALGEBRA. Exercise 140. Find the geometrical mean between : 1. 7 and 252; a^h and ah^-, f and |i ; | and ||. 2. yV '^"cl jJ^o ; 4^:2 - 12a; + 9 and ^x^-\-12x^- 4. Insert : 3. 2 geometrical means between 5 and 320. 4. 2 geometrical means between 1 and \. 5. 3 geometrical means between 100 and 2J|. 6. 6 geometrical means between 14 and — -^^. 7. 7 geometrical means between 2 and 13,122. 8. Which is the greater, and how much greater, the arithmetical or geometrical mean between 1 and \. 9. Find two numbers whose sum is 10, and whose geo- metrical mean is 4. HARMONICAL PROGRESSION. 160. The expressions h ^, \, \, ...., and 4, -f, - f, -4, ...., are called harmonical progressions or series, because their reciprocals 1, 3, 5, 7, , and ^, — |, — |, — |., . form arithmetical series. The general form for such a series is, I- ^> Wh^ ^TT^^ry-.- Hence, An Harmonical Progression is a series the reciprocals of whose terms form an arithmetical series. HARMONICAL PROGRESSION. 385 Notes: 1. Evidently all questions relating to harmonical pro^^ression are readily solved by writing the reciprocals of the temis so as to form an arith- metical series. 2. There is no general formula for finding the sum of the terras of a har- monical series. 3. The term harmonical is derived from the fact that musical strings of •qual thickness and teiisiou produce harvumy when sounded together, if their lengths are represented by the reciprocals of the series of natural numbers; that is, by the series 1, J, J, i, J, J, etc. Harmonical properties are also interesting because of their importance in geometry. Example 1. Find the mth term of the series 3, IJ, 1, f , f , etc. Solution. Taking the reciprocals of the terms, we have ^, |, 1, |, |, etc. ; an arithmetical series. Here, a = |, rf = J, and n = m. Substituting in (i) (Art. 154), d = ^ + {m - I) ^ = ^. Taking the reciprocal of this value for the required term, we have — . Example 2. The 12th term is |, and the 19th term is ^. Find the series. Process. The 12th and 19th terms of the corresponding arith- metical series are 5 and ^. From (i) (Art. 154), 5 = o + 11 rf, ^ = a + 18 d. Solving for a and rf, a = | and rf = J. The arithmetical series is, |, |, 2, t, |, 3, Y> • • • • The harmonical series is, |, |, ^, ^, |, J, i%, 161. A Harmonical Mean is the middle term of three numbers in harmonical series. If a and b represent two numbers, and H their harmonical mean, the corresponding arithmetical series is -, ^, ^. The common dif- .11 !!,«,. ference is ^ ~ o* ^^ 6 ~ H' ^*^^'^'®^» D- — - = T - ^. Hen'ce (solve for //), 25 386 ELEMENTS OF ALGEBRA. To Find the Harmonical Mean Between two Numbers. H - ^^^ (i) Example L Find the harmonical mean between : (1) ^ and ^ ; (2) X -j- y and x — y. Process. (1) Here, a = \ and b — ^j^. Substitute in (i), H = \ (2) Here, a = x -\- y and b = x — y. X^ - 7/2 Substitute in (i), H— '— . Example 2. Insert three harmonical means between f and ^^. Process. The terms of the corresponding arithmetical series are I and J^. Here, a = |, I = ^^, and m = 3. Substitute in (v) (Art. 156), d = ^. The three arithmetical means are ^, ^, ^. The required harmonical means are y\, f , ^. Exercise 141. 1. Find the 8th term of IJ, l^f 2-^2^, .... 2. Find the 21st term of 21 llf, 1^^, .... 3. The 39th term is y^-, and the 54th term is ^. Find the series. 4. The 2d term is 2, and the 31st term is ^*y. Find the first six terms. Insert : 5. One harmonical mean between 1 and 13. 6. 3 harmonical means between 2f and 12. HARMONICAL PROGRESSION. 387 7. 4 harmoiiical iiu'iiii.s buLu uuii | iind ^. 8. 6 harinouical means between 3 and ^. 9. The arithmetical mean of two numbers is 9, and the harmonical mean is 8. Find the numbers. 10. The difference of the arithmetical and harmonical means between two numbers is 1. Find the numbers; one being three times the other. 11. Find two numbers such that the sum of their arith- metical, geometrical, and harmonical means is 9|, and the product of these means is 27. 12. The arithmetical mean between two numbers ex- ceeds the geometrical by 2^, and the geometrical exceeds the harmonical by 2. Find the numbers. 13. The sum of three terms of a harmonical series is 37, and the sum of their squares is 469. Find the numbers. 14. The sum of three consecutive terms in harmonical series is 1^, and the first term is J. Find the numbers. 15. Arrange the aritlimetical, geometrical, and harmoni- cal means between two numbers a and h in order of magnitude. 16. If 50 potatoes are placed in a line 3 feet from each other, and the first is 3 feet from a basket, how far will a person travel, starting from the basket, to gather them up singly, and return with each to the basket ? 17. There are four numbers in geometrical progression, the first of which is less than the fourth by 21, and the difference of the extremes divided by the difference of the means is equal to 3 J. Find the numbers. 388 ELEMENTS OF ALGEBRA. CHAPTEK XXVII. RATIO AND PROPORTION. 162. The Ratio of two numbers is their relative magni- tude, and is expressed by the fraction of which the first is the numerator and the second the denominator. Thus, the ratio of 10 to 5 is expressed by the fraction ^-^ ; the ratio of I to I is expressed by the fraction f -r f (= y%). The ratio of two quantities of the same kind is equal to the ratio of the two numbers by which they are expressed. Thus, the ratio of $5 to $6 is | ; of 15 apples to 3 apples is i/ ; of 3f feet to 5^ feet is 3f ~ 5| = ^|. , The Sign of ratio is the colon :, -^, or the fractional form of indicating division. a Thus, the ratio of a to 6 is expressed by a : b, or a -f 6, or t, any one of which may be read "a is to 6/' or "ratio of a to &." The Terms of a ratio are the numbers compared. The first term is called the antecedent, the second the conse- quent, and the two terms together are called a couplet. A ratio is called a ratio of greater inequality, of less inequality, or of equality, according as tlie antecedent is greater than, less than, or equal to, the consequent. An Inverse Ratio is one in which the terms are inter- changed ; as, the ratio of 7 : 8 is the inverse of the ratio 8:7. A Compound Ratio is the product of two or more simple ratios; as, the compound ratio 2 : 3, 5 : 4, 15 : 6, is 150 : 72. RATIO AND PROPORTION. 389 NotM : 1. A quantity may be detiaed as a definite portion of any magni- tude. Thus, any definite number of dollars, poimds, bushels, acres, feet, yards, or miles, is a quantity. 2. To compai*e two quantities they must be expressed iu terms of the same unit. Thus, the ratio of 2 rods to 9 inches is expressed by the fraction, 16J X 2 X 12 396 163. Evidently the ratios 4 : 5, 8 : 10, ^ : Yi are equal to each other. Ill general, a ma „ I. If the terms of a ratio are multiplied or divided by the sanu number, the value of the ratio is not changed. The ratio 9 : 7 is compared with the ratio 4 : 3 by comparing ^ and |. ^ = ^\, and | = ^|. Therefore, 4 : 3 is greater than 9 • 7. Hence, II. Ratios are compared by comparing the fractions that represent them. If to each term of the ratio 5 : 4 we add 16, the new ratio, 21 : 20, is less than the ratio 5 : 4, because \ is greater than f ^. If to each tenn of the ratio 4 : 5 we add 16, the new ratio, 20 : 21, is greater than the ratio 4 : 5. Hence, III. A ratio of greater iiuquality is diminished, and a ratio of less inequality is increased, by adding the same number to both its terms. If from each term of the ratio 32 : 30 we subtract 24, the new ratio, 8 : 6, is greater than the ratio 32 : 30. If from each term of the ratio 28 : 30 we subtract 16, the new ratio, 13 : 15, is less than the ratio 28 : 30. Hence, IV. A ratio of greater inequality is increased, and a ratio of less inequality is diminished, by taking the same number from both terms. 390 ELEMENTS OF ALGEBRA. a c e g Suppose ^ = ^=^=^ = r. Simplify, br — a, dr = c, fr — e^ hr = g. Add the corresponding members and factor the result, {b-\-d+f+h)r = a + c + e-\-g. Therefore, ^ = 6 + ^ t/t I = ^ = "^ =}" f' H"^'"» V. In a series of equal ratios, the sum of the antece- dents divided hy the sum of the consequents is equal to any antecedent divided by its consequent. Notes : 1. The sign : , is an exact equivalent for the sign of division ; and is a modification of -r . 2. A Duplicate Batio is the ratio of the squares ; a Triplicate, of the cubes ; a Subduplicate, of the square roots ; a Subtriplicate, of the cube roots of two numbers. Thus, a^ : b^ ; a^ : b^ ; Va : Vb; fa : Vb are respectively fhe duplicate, triplicate, subduplicate, and subtriplicate ratios of a to b. Example 1. Find the ratio compounded of the duplicate ratio of 2 a a2 _ -T- : T2 V 6, and the ratio 3ax : 2by. Process, ihe duplicate ratio oi -7- : Tg yB is -r^- : -r^ • , . 4a2 6a< I2a^x ]2a*by Ihe compound ratio -vg- : -p- ? ^ax : 2by,is — p— : 74 — • I2a^x \2a^by ISa^x 12 a^hy bx But — To — : TT— ^ = — To i u — = — = bx : ay. b^ 6* b^ b^ ay ^ Example 2 If 15 (2 2-2 — y'^) — ^ xy, find the ratio x : y. Process. From the given equation, a:^ — jV a: ?/ = ^ 2/^. Complete the square and solve for x, a; = ^ ?/, or — f i/* X Therefore, - = a or - 1 . Exercise 142. Find the ratio compounded of : 1. The ratio 2 a : 3 &, and the duplicate ratio oi^b'^'.a'h. 2. The subduplicate ratio of 64 : 9, and the ratio 27 : 56. RATIO. 391 3. The duplicate ratio of 4 : 15, and the triplicate ratio of 5 : 2. 4. 1 — 3^ : 1 + y, X — X i/ : I + s^, and 1 : x — x^. a-\-h g'+fe^ {a^-l^f a2_9^^2Q , a8-13a + 42 Simplify each of the ratios : 6. 5ax:4:x; li5xy:20a^; 2x^y:\x^. 7. iaxy.z^ay^; -^ ^ :a^na^. Arrange the following ratios in order of magnitude : 8. 5 : 6, 7 : 8, 41 : 48, and 31 : 36. 9. a ^ b : a -\- bf and a^ — I? : a^ + b^, when a > 6. 10. For what value of x will the ratio 15 + a::17 + ic be equal to the ratio 1:2? 11. Find X \ yy if a:^ + 6 ?/2 = 5 a; y. 12. Find the ratio of x to y, if the ratio 4a; + 5y : 3a;— y is equal to 2. 13. What number must be added to each term of the ratio a : h, that it may become equal to the ratio m : nl 14. What number must be subtracted from the conse- quent of the ratio a : b, that it may become equal to the ratio m : 7l1 15. A certain ratio will be equal to 2 : 3, if 2 be added to each of its terms; and it will be equal to 1 : 2, if 1 be subtracted from each of its terms. Find the ratio. 392 ELEMENTS OP ALGEBRA. 16. If a : 6 be in the duplicate ratio oi a -\- x : h + x, find X. 17. Show that a duplicate ratio is greater or less than its simple ratio, according as it is a ratio of greater or less inequality. PROPORTION. 164. A Proportion is an equality of ratios. Four num- bers are in proportion, when the first divided by the second is equal to the third divided by the fourth. a c Thus, if T = -7 , then a, h, c, d, are called proportionals, or are said to be in proportion, and they may be written in either of the forms : a : b :: c : d, read, "a is to & as c is to d! ;" or a : b = c : d, read, "the rat'o of a to & is equal to the ratio of c to d;" a c ^^ b=d^ read, "a divided by b equals c divided by d." The Terms of a proportion are the four numbers com- pared. The first and third terms are called the antecedents, the second and fourth terms, the consequents; the first and fourth terms are called the extremes, the second and third terms, the means. Thus, in the above proportion, a and c are the antecedents, b and d the consequents, a and d the extremes, b and c the means. Note 1. The algebraic test of a proportion is that the two fractions which represent the ratios shall he equal. PROPORTION. Let a:b :: c :d. By definition, a c ~b^d' Free from fractions, ad = bc. Hence, 393 I. In any proportion the product of the extremes is equal to the product of the means. Note 2. If any three terms in a proportion are given, the fourth may be found from the relation that the product of the extremes is equal to the product of the means. Let ad = be. \ a c Divide by 6 rf, b~d' By definition, a:b::c:d. Hence, IL If tJu product of tivo numbers is equal to the pro- duct of two others, either two may he made tlie extremes of a proportion and the other two the 7neans. A Mean Proportional is a number used for both means of a proportion ; as, h, in the proportion a :b ::h : c. A Third Proportional is the fourth term of a proportion in which the means are equal; as, c, in the proportion a : h :: h : c Ia'I a : b :: b : c. Therefore!., b^ = ac. Extract the square root, b = \/a c. Hence, IIL A mean proportional between two numbers is eqv^ to the square root of their product. Let a :b :: c : d. Therefore 1., ail — he, a b Divide by c rf, ~ ~ d' By definition, a. cy.b . d. Hence, 394 ELEMENTS OF ALGEBRA. IV. If four numhers are in proportion, they will he in proportion hy alternation; that is, the first will he to (he third, as the second is to the fourth. Let a :b :: c : d. Then I., bc = ad. Divide by a c, -::=_. •^ a c By definition, b : a :: d : c. Hence, V. If four numhers are in proportion, they will he in proportion hy inversion; that is, the second will he to the first as the fourth is to the third. Let a:b::c:d. By definition, h~ d' a c Add 1 to each member, t+ 1 = ^ + 1, a + b c-^d b ~ d Therefore, a + b : b :: c + d : d. Hence, VI. If four numhers are in proportion, they will he in proportion hy composition; that is, the sum of the first two will he to the second as the sum of the last two is to the fourth. Let a : b :: c : d. . a c By definition, J ~ d' Subtract 1 from each member, a c a~b c— d «^ -~r = ~d"- Therefore, a — b:b::c-d:d. Hence PROPORTION. 395 VII. Jf four niimhers are in propoi'tion, they will he in proportion hy division ; that is, the difference of the first two will be to the second as the difference of the last two is to the fourth. Let a :l ::c '.d. a 4-h c + d Then VI., =-| ^ • also VII., c a — b c — d Divide, b c a+b c+d a — b c — d By definition, a + b : a — b :: c + d : c — d. Hence, VIII. If four numbers are in proportion, they will he in proportion hy composition ajid division ; tliat is, the sum of the first two will he to their differeiue as the sum of the last two is to their difference. Let a :b'.:c :d, e:f::g:h, k : I :: m : n. T»,/... ^ c e g k m By definition, l = d'f=h'l=n' Multiply the corresponding members of the equations together, aek c gm Ff'l'^dir^' By definition, a e k : b f I :: c g m : d h n. Hence, IX. The products of the corresponding terms of two or more proportions are in proportion. Let a:b :: c : d. d c By definition, h~ d' Raise each member to the nth power, fcS ~ ^ * 396 ELEMENTS OP ALGEBRA. Therelore, a" : 6" : : c" : d\ 1 1 Extract the nth root of each member, — = -^ • Therefore, a" : 6^ :: c" : cT. Hence, X. /?i a^iy proportion like 'powers or like roots of the terms are in proportion, A Continued Proportion is a series of equal ratios ; As, 8:4::12:6::10:5 ::16:8; a : b :: c : d :: e :f::g:h, read "a is to 6 as c is to d a.s e is to/ as g is to A." Kote 3. Four numbers are said to foiin a continued proportion when each consequent is the antecedent of the next ratio ; as, a : b :: b : c :: c : d. Let a :b :: c : d :: e :/ :: g :h. a c e g By definition, ^ = ^^ = ^=: ^^ • a -^c -^ e + q a c e g ByV., (Art. 163), l^d-v f+h^ h^ d = r'h Therefore, a-\-c-\-€ + g'.h + d +f -hh i: a : b. Hence, XI. In a continued proportion the sum of the ante- cedents is to the sum of the consequents as any antecedent is to its consequent. ^^ a2+62 ab + bc Example 1. If - .— r i> ^ = ~"a2~i — 2 » prove that 6 is a mean ab + be 6^ 4- c^ ' '■ proportional between a and c. Proof. Free the given equation from fractions, transpose and factor, (b^ — acy — 0, or b^ = ac. Therefore II., a : h :: b : c. Example 2. If a . b :: c : d, prove that m a'^ +^p b'^+ nab : mc^ -i-pd^ + ncd :: b^ : d^. iVI., a b c~ d a« ab 7^=7d' a* b» ab nab a^ n (t li cd- ncd' or c2 ~ 'nc il b^ pb^ a^ pb^ d^' ~ pd^' or c^~ pd^ PROPORTION. 397 a b ^1 V Proof. From the giveu proportion VI., c~ 7l ^ a Multiply by - , Square Loth members of (1), By I. (Art. 163), Also I. (Art. 163), Also I. (Art. 163), r» = ^a- m a* pb^ _ n ab _ b'^ a' Hence, ^^ =^ = ^^-^ = ^ = -^. By V. (Art. 163), ^...^'pd^^ncd = d^^' Therefore XL, ma^ + pb^ -^-na b :mc^ + pd^+ ncd :: b^ : d^. Example 3. Find x when ^m-\-x-\- ^m-x -. ^m-^x-^m—x ::n ; 1. Procefw. By VIII., 2 ^m-\-x : 2 ^m-x :: n+l : n-1, or I. (Art. 163), /^/^Ta: : ^m=^ :: n+l : n-1, By X., m + x : m -x :: (n+l)» : (n-l)«. By I., {m + x)in- 1)» = (m-x)(n-M)«. Simplify, transpose, and factor, 2 n {n^ + 3) x = 2 m (3 nH 1). _m(3n« + l) Therefore, x - ^(^8^.3) • Exercise 143. J( ad = bc, prove that : 1. d :b :: c : a; d : c ::b : a; h : a :: d : c. 2. hidr.a'.e; c:a::d:h; c:d::a:h. 398 ELEMENTS OF ALGEBRA. Find a mean proportional between : 3. 2 and 8; 3 and 1 J ; Handf; 8 and 18; a^bsmdah^. 4. (a + hf and (a - hf ; 360 a* and 250 d^ h\ Find a third proportional to : 5. I and f ; I and | ; .2 and .4 ; 2 and 3 ; f and f 6. 1 and VI; (a - hf and a^ - 62; ? + ?^ and - • Find a fourth proportional to : 7. 2, 5, and 6 ; 4, |, and | ; f , f , and f ; a, ak, and 6. 8. a3, ah, and 5a22,. ___, _______ and ^g--^ If a : & :: c : 6?, prove that: 9. a + h : a :\ c -\- d : c\ a — h \ a w c — d : c. 10. ac :b d :: c^ : d!^; ab : cd :: a^ : c^. 11. 2a + 3c:3a + 2c::26+3f^:3&+2c?. 12. 3 a - 5 6 : 3 c — 5 c? :: 5 a + 3 6 : 5 c + 3 c^. ^oo A7 o ^7^^ + ^ r^, — 6 a b 13. fa:|?.::|.:H; -^ = __^ . _ = ^ . U. 3 a + 2b : S a - 2b :: 3 c -{- 2d : 3 c - 2 d. 15. la + 77ib : pa + qb :: I c + md : p c + qd.' 16. ft3 . 2,3 .. ^3 . ^3. ^2 . ^2 .. ^2 _ ^2 . ^2 _ ^2 17. a2 + c2 : a& + C6^ :: a6 + C6? : b^ + ^2. 18. V^r:r^:V6::VM^: V^; l = \^P^' PROPORTION. 399 If 6 is a mean proportioual between a and c, prove that; If « : 6 :: c : rf :: 6 :/, prove that : on J. . i 7.i^_L/ O'+'^c-^Se 2a4-3c+4g 20. a:fe::a + . + .:^ + ^+/; ^^2rf^3/ = 26+3rf+4/ ' 21. rf is a third proportional to a and 6, and c is a third proportional to b and a, find a and 6 in terras of d and c. li m + n : m — n :: X -\- y : X — y, prove that : 22. ^ •\' w? \ ^ — iv? \\ y^ ■\- n^ \ y^ ^ n^. Solve the following proportions : 23. 24. a:3 ~ yS : (a; - y)8 :: 19 : 1 and a; : 6 :: 4 : y. 25. If = = , show that a + b-{-c = 0. X — y y — z z — X 26. A and B engage in biisine^Jj with different sums. A gains SloOO, B loses $500, after which A's money is to B's as 3 to 2 ; but had A lost $500 and B gained $1000, then A's money would have been to B's as 5 to 9. Find each man's investment. 27. Show that the geometrical mean is a mean propor- tional between the arithmetical and harmonical means between the two numbers a and b, 28. When «, &, c, are in harmonical progression, show that a:c::a — 5:& — c. Hence, of three consecutive terms of a harmonical series, the first is to the third as tht first minus the second is to the second minm the thirds 400 ELEMENTS OF ALGEBRA. 29. Find the ratio compounded of the ratio 3 « : 4 6, and the subduplicate ratio of 16 &* : 9 a* 30. If - = 3i, find the value of ^ ~ '^ ^ - y ' 2x- by 31. If 6 : a : : 2 : 5, find the value of 2a-Zh:Zh—a. 32. If T = T, and - = ^, find the value of ^7 ^^ • 6 4 2/ ' 4:hy — lax 33. If 7 m — 4 71 : 3 wt + ii : : 5 : 13, find the ratio m : n. 34 If s s— = -TT , find the ratio w : n. m2 + 7l2 41 35. If 2 a: : 3 y be in the duplicate ratio of 2 a? — m : 3 ?/ — m, find the value of m. a c m 36. If - = - = -, prove that each of these ratios is equal to ^WHH^L 4 m^c 37. If 2a + 3Z» : 2,a - 36 :: 2^2+ 37i2 : 2^2- 3ii2, show that a has to h the duplicate ratio that m has to 7i. 38. A railway passenger observes that a train passes him, moving in the opposite direction, in 30 seconds ; but moving in the same direction with him, it passes him in 90 seconds. Compare the rates of the two trains. Solve the following proportions : 39. Vx + Vh : Vx - Vh '.: a :h) 2"^' : 22- ; : 8 : 1. (x + y-.x : X — y '.'. m + n \ m — n* 40. ^ „ o 2 2 . 2 1 APPENDIX, COMPUTATION OF LOGARITHMS. Since the logarithms of all composite numbers are found by add- ing the logarithms of their factors (Art. 122), it is only necessary to compute the logarithms of prime numbei-s. The following method for computing logarithms is the one that was used when our tables were first made, although it is not the most expeditious method now known. Example 1. Find the logarithm of 5. Since 10«> = 1, and 101 = 10 (1) and as 5 lies between 1 and 10, its logarithm must lie between and 1. Extract the square root of (1), 10-6 = 3. 162277+ (2) As 5 lies between 10 and 3.1622774- its logarithm lies between 1 and .5. Multiply (2) and (1) together, I0i» = 31.62277 f. Take the square root, 10 '» = 5.6234134 (3) 5 lies between 3.162277+ and 5 623413+ , and its logarithm must lie between .5 and .75. Multiply (2) and (3) together, lO^-^ = 17.7827895914+. Take the square root, 10«26 = 4.216964+ (4) Since 5 lies between 5.623413+ and 4.2 16964+ , its logarithm must lie between .75 and .625. Multiply (3) and (4) together, take the square root of the result, and we have 1()«876 _ 4.869674+. Continuing thi; process to 22 operations, we have, 10«8»7(h- = 5.0000(K)+. Therefore, log 5.000000+ = .698970+. 402 ELEMENTS OF ALGEBRA. Example 2. Find the logarithm of 2. log 2 = log i^ = log 10 - log 5 = 1 - .698970 = .301030. Examples. Find the logarithm of 11. 101 = 10 (1) • 108 = 1000 (2) Extract the square root of (2), lO^-^ = 31.62277+ (3) Multiply (3) and (1) together, lO^-s = 316.2277+. Take the square root, lO^-^s = 17.78278+ (4) Multiply (4) and (1) together, 102-26 - 177.8278+. Take the square root, 10i-i25 = 13.33521+ (5) Multiply (5) and (1) together, IO2125 _ 133.352I+. Take the square root, 10i-0625 = 11.54782- (6) Multiply (6) and (1) together, 102-0625 _ 115.4782+. Take the square root, 10i-03i25 _ io.74607+ (7) Multiply (7) and (6) together, 102-09375 ^ 1 24.09368+. Take the square root, ioi.o46875 ^ 1 1.13973+ (8) Multiply (8) and (7) together, 102-078125 ^ 1 19.70845+. Take the square root, 10i-o390626 _ 10.94113+. Therefore, log 10.94113+ = 1.0390625. Continuing the process, the logarithm of 11 maybe found with sufficient accuracy. Example 4. Find the logarithm of 3. Take 10^ = 1 and lO-^ = 3.162277+, and proceed as before to 14 operations, and we have log 3.0000+ = .47712+. A table of logarithms to four decimal places will serve for many practical purposes. In the tables most generally used by computers they are given to six places of decimals. Seven to ten place loga- rithms are necessary for more accurate astronomical and mathemati- cal calculations. ANSWERS TO THE ELEMENTS OF ALGEBRA BY GEORGE LILLEY, Ph.D., LL.D. EX-PRESIDENT SOUTH DAKOTA AGRICULTURAL COLLEGE 'TEACHERS' EDITION SILVER, BURDETT & COMPANY New Yobk . . . BOSTON . . . Chicago 1894 Copyright, 1893, By Silver, Burdett and Company. John Wilson and Son, Cambridge, U. S. A. ANSWERS TO THE ELEMENTS OF ALGEBRA Exercise 1. 1. a plus 100 ; a plus 10, minus 2 ; etc. 4. q plus t, plus 8 multiplied by m ; etc. 6. m-\-n-\-r~t] etc. 7. m -^ n -\- b\ m + n — b. 10. X — m ^ n marbles. 11. 7i + a; -f y 4- 6 4- w. 12. a-rrti-\-n — k — X — 1/. Exercise 2. 2. A; times ^, plus m times A; divided by c times w, plus a divided by 6; or, to the product of k and /, add the quotient obtained by dividing the product of m and k by the product of c and n, and to this result, add the quotient obtained by dividing ahy b; etc. Exercise 3. 1. xyz; 5mn; Sxy-j 15abmn. ^ ab ab 25 mn r , 2. — j-T, T. 3. ■ , etc. a -\- b a ^ b m -{- n ANSWERS TO THE a mn 4. 10. ^. -^. 20 amn. o o 6. 18 mi'. 18 m 71 + 18 m r. mn 8. aoc -\- mnr, 13. 9. . G 11. 25. ^. 14. 5. ^»' + ^^' m r dt dt dt 15. It. \- bt. r . n n n Exercise 4. 1. m fifth power; 3, m fifth power, x second power; etc. ; a, h second power, plus h ; etc. 3. 10, ah tenth power; m third power, n third power, m n third power, etc. ; the third power of m second power minus 3 n. 4. Etc.; 3, a second power, b, times the third power of a minus b second power ; a second power plus b second power, times the second power of a third power minus b third power. 6. 10 m, plus n fourth power, times the fourth power of 10, n second power, minus m fifth power is less than 15 a times the second power of x, minus y second power, times the third power of x plus y ; etc. 8. m + n. 2x. (a + by. (x — y)\ 5 (x — yf. 9. (x' + yy. (icH- 7/2)2. m^x^y^. ^xy^. {x'' + y"") {x^ - y% 11. 1 m'^n^—2n'm^+^a^h'^+^a''h^ + 5a\ '/a-\-b^ m — n, .'. (a + by = (m — ny. 12. .'.X = m^,'.'x + 3m^ = 2x + 2m\ a + a + a to ?i — 2 terms = (n — 2) a; etc. ELEMENTS OP ALGEBRA. Exercise 5. 1. 3; 448; 60; 180; 64; 9375; 390,625; 1792; etc. 2. 144; 60; 64; 250; 4; 40; etc. 3. T^; 3; 7; 75; 32,400; 288; etc. 4. 6; 60; 2; 3; 5^; 72; 0. 5. 9; 160; 2048; 81; 8| ; 13^; ^; H- 8. 20; 11. 12. 2; 0. 16. 4|§f ; 2^. 9. 0; 12. 13. 249; 134. 17. 1; ||. 10. 14; 6. 14. 22; ^. 18. 5; 58^. 11. 18; -14; 2. 15. ^; 36. 19. i; I4. Exercise 6. 1. 11, negative. 11, positive, etc. 2. is 6 units greater than — 6 ; etc. 3. 6, 3, 9, 12, 11. 10. b-a. 8. 2 times the expression in brackets, 3 i in the posi- tive series plus 5 a in the negative series, and from this result subtract 6 times the expression in brackets, a in the negative series plus b in the positive series ; etc. 10. Value, 26. 11. Value, 1 + (- a:^^) + (+ a;«) + (-x), 12. Value, -2. 19. 80; -624^^. 26. 56; -15. 13. Value, 0. 20. 3; 15^^; 1^. 27. 2; 15. 14. Etc. 14§^§. 21. 9; 2; 15. 28. 102. 15. Etc. 17. 22. 127; 21; 6; 1. 29. 52; -18^^. 16. Etc. 6. 23. 3 ; 6. 30. — 15 ; 12. 17. Etc. ^. 24. — 4 ; 6. 31. 2 a + b. 18. Etc. -2r»^. 25. 16; 55. ANSWERS TO THE 32. 5« + (6 - 1). 33. h + ^±_^. 34. x-\-x-{-x-{-....toa terms, or ax. 35. n, n -{- 1, n -\- 2. 36. m, m — 1, m — 2^ w — 3, m — 4. 37. (a — 1) m ; (m + w) m. 38. X — S, X — 2, X — if X, X -\- 1, X + 2, X -\- 3. 39. a- -2/". ^ , a; 40. 2x + '^ + abc, 41. ^. b y — c 42. a" + - — 5 ( ~\ >h — x. 43. (cc^ — ^»« 4- 2/'") ^ ^" < g'^". a** « — ^> 44. x"" — b"" a' + b^ (A 45. — ^ — .X ?/ + cc + X + cc + • . . • to w terms — a"*. 46. x"* H ■=x — y-\- (4a+^» — m) + — . 47. 5^8 — 3 6i^Z>8 + 2^»2. 49. %x^y'^ — a^b\ 48. 3^2- 2cc«?/ + ^3. 50. + 6a(ir 4- 2/-^). a: — y Exercise 7. 1. (+15.1 a). 5. (-5£c2). 9. (+3.8a-^c(/). 2. (+15 ax). 6. (+/2^). ' 10. -7.71 (^' + c). 3. (+41c). 7. (-21a8). 11. + 5.81 (x - ?/)8. 4. (+ 2 a ^ c). 8. (- 3.5 a" U^). 12. - 3/^ (|) . ELEMENTS OF ALGEBRA. Exercise 8. 1. (-6.69 a;) + (+3.5^). 4. (+2^ x') -\- {+ i ab). 2. (+Ha) + (-5'5«^)- 5. 14.9 a- 2.67 (x-y). 3. (+8a'a:) + (+7c«x'). ^ 6.1Q-2.3Q. Exercise 9. 1. 5 y. 2. 4 // — 5 a. 3. x. ^. a + b -\- 2 x -\- t/. 5. —33 ax — 4:bd -\- 5ni7i. 6. -2a + ^ + c + </ + ^'i + a^. 7. — 8 a i c -j- 53 a 6 m — 20 c m. 8. 4 w. 9. 3 a -h 3 /> + 3 c + 3 rf. 10. 4.25 a^ 4- 3.3 f + 2 6. 11. 2{^^a+l^^ab. 12. iVo ^'^■^ + -'^l ''^ — 2t\ w n — 2^. 13. — ac+1.92frf. 14. — (1*^.5 a^b+ .25 a b^-{-b\ 15. 7.8 (m — ny — 6.03 (« + y)^- 16. a^ 6» + x^y, 17. 7tJ5 a^ + 5|i « a^« + T^^ //« + 3^ x^ U- 18. I «• 4- J\ Z^'' - 1.3 c» - «2^ + m a^c-.2ac^-^l}fab^ + ljgga6c + 2*3c-1.3^c'». 19. 21.33 a^ + 8.37 ar^ - 5 x -f 8.5. 20. 2.6 a* b^ r» + 2.8 «« 6» c^ + 3.91 a^ 6» c*. 21. 6.09 a* + h f' + 4.97g 6 c + 3.5 c"* — 3.03. 22. \aX"l.Sab + 5.2x + ^xy. 23. 12.91 a + 4.1 y — 6.82 z. 24. 3 o 6 + 10 J- y 4- 9 a- // — .T« y + 22. 25. - J(m-3a-)". 26. 5 a*. 29. <?» + ^.« H- c« — 3 a 6 c. 27. 10a« + 86«+12a + 12. 30. 0. 28. x* — y*. 31. 4 a:'" + 2 a" — a". ANSWERS TO THE Exercise 10. 1. lOaHc; -SSab^xi/, 7. ^ (x + y). 2. 2x'y^; 0. a -3ax\ 3. —daxy, xy^ + bc. 9. 26axy\ 4. —3.75x\ 10. %x^y. 5. -2.72 a bc^ 11. 8.9i(a + ^)2. 6. —13.1 VI np^^ 12. 4TV5ra^". Exercise 11. 1. —2x + 4:y — 3z. 5. ~-2x + y — :^z. 2. -2x-y-4.z. 6. -^.T + |2/ + i 3. —2a — 10c. 7. 2a^ + 3aH — 3hc—10, ^. Ix+^y + l^z. 8. 2x^y + 2abx + l. 9. abcxy — 3aby-}-5bx — 3. 10. 2^372/ — 5ac?/ — 2a^c + a + l. 11. .6 ic* - 1.8 x^ + 7.03 cc - 9. 12. —x^ — 1.2x* — 2x^ — 2x-\- 2. 13. ^^ 77Z8 — I y«, 7^2 _ J s ^2 24. 7^8 — «. 15. ^^m^--J^y + 2lin-^x. 16. 2.5«2^,c — l^a-?/"- 4t^c. 17. 0. 18. 2 a;*" + 4 ;?;" y"* + y"^. 19. 1.8 ^/"^ x^ - 1.7 a ^»3 .X + .37 b"" c x"" + .7. 20. a^ _,_ 3 53 _ g ^4^ 21. 3.55 ic^ + 33. 2y^x + S.5x z\ 22. ic'» + 2y"'. 23. | iy _ 1.9§ 2/^ + 1-7 ?/« + 3. 3. 2-2a^-2a-2. 4. 2 a ^ — ^2 _ ^2^ ^2 n yin rjf.Zn 2 ^*" Exercise 12. 1. — 4 m^ — 2 w - - 5 7z2 + 6. 2. 4.x^-Uy. 5. — 2 m. 6. 4?/ w-l ELEMENTS OF ALGEBRA. 9 7. -xhyi-ei/^ + Sz' -^1. 9. - ^ ari + 2.9 a;* yi - 33 yl 10. 3.9 cyi + 1.6i ax + .31 ^» - 1.2 7m - 5^%. 11. 5.5 6" — 3.25 «"• — 4 w. 12. y- — x^ — 1.2 i". 13. ^ c' — l.Ta'" — o^b"* — 2 d". 14. c» - i (a» - by - 12 (x» + y2^«. 15. 2 a^ 21. 6 rt- — 2 a ^. 16. 3 a;«* + 2 x^* — 6 a;". 22. aJ ^i. 17. 2^2;— I a^- 2. 23. 4y a^- ^a» — 4.5a + 3f. 18. \ //i« +1. 2^. llbc -\-l mn — ^xy. 19. 7 J/'' + 18y — 4. 25. 4 x*-^ — 2x'''-^. 20. 0. 26. x"— 4. 27. 64 (x 4- y)* + .3 a- + x"* - 3^ a-x" - C + 3. Exercise 13. 1. 7x8; 15a«x«; 2a'*^*x«. 2. ix"y»«m««"; 2an»c«<;»w»w». 3. a6»c»y"^*; ?a"x»y"«. 4. 30«"x*y"; ix'" + *y"+^ 5. 2a-^'/>^V^*»x^<>y"; Ti5a'Z»"r"j'" + »y"+». 6. vSa"+>6'' + ''x^+V"^*; ^'/A 7. a'" + -6»+»'; 5.7 x*y". 8. a»6»x«y«; a^-ft^". 10. 2a»x»y". 15. .765a«mM+'x— "-'1. 11. 3a*'+''6" ♦c'+'"rf*. 16. l|a'"+*'-**^-"//"+*«'". 12. 2a* + »c*+-+*rf«x'"+". 17. l\(a-\-hy\ 13. 2 a*l m^ n** x*» /. 18. 3 (c + ^)><> (a + ^)". 14. 5 aHi a;*A yi. 19. ^ (« -|- ^,)-+ *» (x — y)'"+«. 10 ANSWERS TO THE Exercise 14. 1. —2S10anG', a^x^'f. 6. -3a-H-^c^^d. 2. w'hx^y'; —ha^'^hU^d}. 7. an-^(Px\ 3. 4:aHcxy) .Sacm^x^T/^ 8. — .3 a^^ x^ yi -^ aH x^. 4. ^a^bcdx^yz; a^^+H'+^x^"" y\ 5. — 3 a^b^x^yzvw, a'«+" + i^>" + 'a:"* + ^ ?/"+\ 9. — m^ n x^t yH '^ — 2.1 ai bl 11. A X 3" mx^ y'^^'p q"". 10. a^'^ ''■ a aa--- -to 10 ni3iGtovs. 12. _ 2^« + »"3^°« + ^" + i. Exercise 15. 1. abU^+aHc^-aH'c, ^ aH'* - :^ a^^^ c^ - ^ aH^^ e^ 2. 5 a^ Z*» c^o — a^ b'' c"" — 2a^b^ &'' ; .12 ic^ y^ - .1 £c^ / 3. 3m^ n — 1\ w? ii^ + 3 m ?i^ ; 9?- y — xy'^ —l\x^ y^. 5. 9 a«Z»ar3— |«H^aji3^ .3a^>«ar«; p^ x'^'^r — p qrx'^-^'' — pr^ ajtn 6. 3«'"^>2_2«^« + 4 rt»» + l^n + 2. .Ga^'^Z;^^ - 2 6/"'-^^>2p 8. a^i ^^s - a^^ ^»^^ - a^ b^^ + a^b; x^ y-h - x^ y\ -\- x^ y — .Bx^yl 9.-2 .T.3 ?/^ + 8 a^2y6 _ 8 _^ yv . ^„2 ,^o _ 2 ^^p ^2? ^ ^o ^^2,^ 10. ^V «^ ^'^ ^^ + ^7 «* ^ cc' + ^ a^ Z*^ ic^ 11. — cc§ ?/ + .-^^ ?/^ ; ai ^'«^ — 6?i b'^\ 12. 45a!i3i7/°i-45£c'«^/3>. 13. ^V ^'^^'' — :\ ^'''^"■•'^'^' — -2 ^>'3\tV 4- ^ //7-5. 14. 180 a-Sm/.-am _ 189 ^-4m ^--im _ |4Q ^-2m Jl-2m^ ELEMENTS OF ALGEBRA. 11 Exercise 16. 1. a*-{-aH^+b*; a* + 4 a^x^ + 16 x*. 2. x^ -\- i/; ^xij. 3. 2/' + -'y'~lly'-12y + 27; y* + a y» + « V — a t/ z — a*z — a*. 4. \x*-l,^^x' + ^jf', .64 a» 4- 3.24 a ^2 - 2.7 ^»». 5. a^-{-2x'-2x'i/ + x^-xi/- 7/ -{•!/', l^x*- 1^ ax« + i a^x^ — I a\ 6. a;« + 8a:y + y — 1; H a:^ + 6 aic* + f a*x« + faa^* + 3a^x^ -{- la^x _ ^ a«x« — ^ a«a- - j^jj a». 7. a-' - 7 a;« + 21 a;* - 17 a-* - 25 a-« 4- 6 ^2 - 2 a; - 4. 8. a» + 2^i«4-5a* + 2a24. 1. 9. x» -h 1.25a;' + .25a;« + Sx*^ + .5.r* + .25 x' + 1.25 a:^ + 1. 10. 4 + 32 a - 4 a^ -f 25 a» - 6 a* 4- «». 11. x» — 32 y*. 13. a« — 3 a ^> c + // 4- c». 12. x^* 4- y^^ \ x^ -\- i/. 14. «» 4- 3 rt ^» c 4- ^' — c*. 15. a*^»2 - rt^-2 4- ^*<^ 4- ^f/^. 16. — 8 a-2 //. 17. .3 m« 4- 2.90 « w — 3.01 i m - .3 71-^ 4- 3.01 a 7i - 2.99 b n — .01 a^ 4- .1 ^. 18. a^x^'* + b^x'^'* + 1^ -\- 2a^»a;'" + " + 2ara:" + 26rx"; ari — yl 19. a*" + 2a'" 6- 4- b^"', a*"* — i*-; a:l 4- &^*+ J5a-*4- ^'. 20. 6a;'" — 4a;;/'-*~9ar— V* + 6y; a^a-m+a _^ ^^^,3.n+2 + a^6x* — a^ar'"+' — i'^x-^'^-ff^^'^a:* — ax"" — ^x" — a Ax. 21. 3a*'"'x4-3a2 + -y + f(2''^'" — Sri^'^+^a; — 3a'-*-''?/ — a'" 4-3a«-x2 4-3a«xy 4- «*"^; x5 — x=//-? — xiy + y*. 22. .04a — .09 61; x — y. 24. l + x + x^ + x^ + x^ — x^ 23. X* — i/-\ 25. X* — 5 a* x* 4- 4 a*. 12 ANSWERS TO THE 26. 120 ic* - 346 x^ - 205 x^ + 146 a; - 120. 27. x^ + x^ + 1. 28. x^ + a*x^+a\ 30. a^"*— Z.6«, 29. 3a^-oa'b-SaH''-j-7 aH^ + 6aH^-2ah^- h\ Exercise 17. 1. a2^2a-15; &2_^^>-30; a;2^7a; + 12; a;2-3a;-4 a;2 - 5 £c - 14. 2. ^2— 14ic + 48; a2+4a-45; a2_4^_32. 4a;2-18a;+20 9ic''^+ 6aj — 35. 3. :c6 — 7ic^2/^ + 12?/^; a;^ + xy — ^y'^; a^"^ + a"" — 2 9:i;i«-27a;5 + 20. 4. 4a*/-8ay-32; 9 a^ a;2_^ 9 a ic - 28 ; a;« - a a^^-12a2 • x'o _ ^2 ^5 _ 6 ^-2 5. 4a;2_2(ia;— 2^2; 4x2" + 4 a x*^ — 15 a^; 9^:2 — 60:3/ — 2 y^; Ax^ — Amx^ — 24: rn}. 6. a;2_6^^_|_5^2. ^2^3 ^ ^_4q^,2. ei^ _- 8 a^ £c + 12 a;^ 5 25x2»-5a2;K^o_i2a^ 7. 25 a:^« - 25 cc5^2 ^ 6 2/^ 9 a^« + 3 a^ (2 a ft - 4 a ^/) -%a'b'; «2" + a" (3 - 6) - 3 Z». 8. 16^2 + 4a(^>- c) -^»c; 2na' — 10a(b^c) + 4.hc', a^y'^ + I aa;^/ + ii^^'-, «^ + i «' — I- 9. 4ar + 26 a;^ + 12; 4 a + 2 a^ (5 — 3 a a;) — 3a ^»a;; .09 a; 2/2" - .3 a;^ ?/" (a; + 2/) + «^ V- Exercise 18. 1. 4a;^-92/^ a:^ - 4 2/^ 25- 9x^ 25 a:'^ - 121. 2. 4 a;2 - 1 ; 4 a;2 - 25 ; 25 x^y' - 9 ; c^ - a^. 3. c* — a^\ m^ri^ — 1 ; a^ y^ — }p-\ a^ x^ — 1. 4. x^ — y^; 1—p^q^; m^ — n^\ a^"* — a^". 5. 25x2?/-2-16 2/^ 25a!^-9y^ x^-9x\ ELEMENTS OF ALGEBRA. 13 6. ^a'^x' — b'^y^', m-^^ — ii-^^-, 100 a" 2"' _ 169 6"''*. 7. m + w; 16a — 400a;*; al — b-%, 8. 121 a:- 900 y; 225 a* *•- 256 a 61. 9. \aH-^-tb-^x-^; a^-b\ 10. aH^-l\ 16 a*'" — 256 a*". 11. 625 a»2 - 1296 b^ ; a" ^^ _ «« 6". 12. x-\-x-'y^; |ec-'"-^af^"*- Exercise 19. 1. 4 a** — a-" + 6 a" — 9 ; 20 a^ — 3 x2« — a;'- — 6 x^**. 2. 4 a*' 4- li a" + 3§ a^^ + 6 a' - 15 ; 2 « + ^5 a;5 - | a;4 3. _ § a» + 6^ at — § a2 — 4^ al + 6 a* — a« + I a-l - a-2 ; 6 a*"—'*' + 3 ar--^b^f — 2 a^-sp^-sp _ ^p ^-p 4. 25 a^V* + 20 a;2«2/26 _ 15 a:« y* _ 12 x« y% a^* + a" + a— 4- a-*". 5. .09a«-.156a*6 + .22a*6«-.488a»6«-1.39a«6*+.3a6* + .rib\ 6. 1 _2a;J-3a;i + 2aji + 2a;i; a\ — al ^- 4|al-8ia-i + a-i - 2 a-i + i a-» - i a'^ + «-« - 2 a"*. 7. -2a;t + 2x5-3x-5a;-l + a;-84- 3a;-«; al"+a-i". 8. x""+' — 2a:*"+'*' — 2ar2''+2 4.2a^'' + ' + a:''^* + 2 3f + '— a^+* - 2x* + x"-'; j;2n+2_4^2n _^ 12a:2--i — 9x''*-*. 9. 2a:»"-*-4a;"'-' + 2.1ar«»-.9ar*''-'+.lx*"+2+.2a;'»+* 4. 2a;»"-» — Sar*— 2 + x«— » _ ar^-. 10. 9af+"-' — 34a^+— » 4- 29 ar + - + » — «•"+"+*. 11. 2a;^+> — 63:*+* 4- 2x«+'^ — 4a:'+* + 3a:*«+» - 9 a:»«+* + 3a:*^+» — ^Q^'^''-^ — 4a:»'^+» + 12x«'+* - 4 a;'«+» + 8a^'+^ 12. ITar^-^V*"^*— lOx^^+V + l^x^'-^V"^'— ^^"""V*"^* 14 ANSWERS TO THE 13. m^ + '-— 3m^+'-i 7i+ m^+'-^Ti^— m'+'-^Ti^ _ 3^p+r+i ^ + 9 mP + '- n^ — 3 mP+'-'^ n^ + 3mP + '■ -'^71^ -{- 7iiP-^'-+'' 71^ 15. 2 x^y — 18 x-^^^ + 6 a;3y _ 18 a:'^ / + 4 a:?/^ 16. f — x-^"*; ^^x'-y-'^b _^^^-'zay2i,^ 17. x^"— 2/2m. ^_^^_20; 49x2 — 9^-2, 18. 16 x2 — 8 x-0 - 15 a:-2 ; ^cH-^ — ^\ ^» Z/"* ; a^n _^ ^14 ^n + 49-9a-2«. Exercise 20. 1. 16aH^', 21a>^m', 32x^y^', .00032 a^o^^i^c'-O; .01a2»^,2«. 2. 49a«^^ 121 a2&4c«<^^; - 27 c^ a:« 5/I2 ^i^ . 27a^^^»«y^ 25 a^^^^^^yo ^20^ 3. 256 a« 6« ci« ^^^4 ^^32. ^10 ^10 ^10 ^10 ^10. _ ^^9 ^e ^s . ^8 js. 729a^6^8c^; -^aH\ ^4nj.3n^«^2n^ 5. — 8a-^'7»'«"; m"=; ««'; a^b-'^c-^^', m""*'/!-"*'"; -8; a^n^ 1. 6. 16a*5-«c*?^^; ri^^ + i^smn^ ^»»« + ia-m. ^rri'n^x^^f. 7. -54am3 7i»a:^2. 81 a^« 5«^ c^* m^^ ; 81 a^^,-*. a«^-« + i. -243a^-20c-'#ic*?/§. 9. _ ^14^21 ^56^28^. _a^rm,/»5 ^-'"'b^'k^*; _8a2i^,i4. x-^y^\ 10. X^2«^42m. ^21^65. 4" ^7n J7n ^7»« . _^7m»n«. J^Uk ^Ukm^ 11. m" (a — 3 <^)P" (x — 7/)'" ; (a — 3 c^)i"' (a; — 2/)^" ; 3" (a — & + c + c?)" (a — cc)""*. 12. a" ^>" c" (a — &)"*"(cc + y + z'^Y" ; ct"' (x — y + ^)^"' (cc — y"»)8'»*. ELEMENTS OF ALGEBRA. 16 Exercise 21. 1. a;« + 4 X + 4 ; m^ -\- 10 m + 25 ; n^ + 14 w + 49 ; a«-20tt + 100; 4a;* + 12a;y + 9y^ a"" -\- 6ab + 9b^; 2. J-2+ 10xy4-25y2; 9x«-30a:?/ + 25 2/'^; 4^2 + 4^26 + a-62; 25x2-30x^^ + 9x2^2; 25a2^,2c'2— lOaic^ + c*; a;iy-i_4icy + 4?/*; a^m ^ Qa'^b-" -{- ^b-'^\ 3. 4 x^ 4- 12 a* X + 9 a ; x^ f/^ + 2 x^ y -{- x-t -, 9 a-* + 30a»6-» + 25ai»^»-«; l-2x + x2. l_2cy+cV; m2-2w + l; a2^,4_2a^-^+l; || a^-- /^ a* + ^i^. 4. ^a2^-*-f 3a6-»x-» + |6-2x-2; j^*'*^*'- — 2j9*^'"r«4- r^'; .00000004x2'" + .000002 x*"//" + .000025 y2». 5. 3»j/wMtV — 2w»7i'' + 'y/ + Y^^''^^"'; icV + 2y2a;« + //2^2_^3.a-g2 _j_ 2ar2y^ -I- 2xy«2. 4x* + 5x«+l + 12x»-6x; x* + 6x2+ l_4x» — 4x; x*-4x2 + 16 + 4 x» - 16 X. 6. 4a?*H-13x« + 9-4x»-6x; x« + 25x* + 4 - lOx* — 4 X* — 20 X ; 16 w* 4- w* ^2 + 7i» 4- 8 m^ n* — Sn* — 2m^n^\ x» + 9x2 + 4__6x« + 4x* — 12x. 7. x2 y2 _|_ 4 ^8 ^ 1 _ 4 ,j jp y _l_ 2 a; y — 4 ?i ; rti^ ■\- v} ■\- j^ -{- q^ — 2 m7i — 2mp — 2mq-\- 2np -\- 2nq -\- 2pq] a;« + 8x^ + 16x2+ 9-4x'^-14x^-12x; 1 + 3x2 + 3x* + x« + 2x + 4x« + 2x»; x2 + 9^/2+ 4^2 + ^2 + 6xy + 4ax — 2/>x + 12 ay — 6 bi/ — 4ab. 8. 4x*+5x*- 17x2 + 9 + 18x»-6x; x2+4y2^9^2 + 4n2 — 4xy— 6x2 + ^ n x -\- 12 yz -\- ^ n y — 12 nz\ ^2" + n2"* + />2» ^ ,^2« ^ 2 ?;^'•7^"• + 2??^''/>" — 2m* q"^ + 2n'-/)» — 2n-^'" — 2/>»^'"; ^a^ + g^ + f — 2a6 — a + 9 ft. 9. 4a« + 4 62+TVc2~2ai + iac-Ac; x2" + y2m _,. ^ ^2 + i 62 _ 2 X* 2r + <» X- - 2 6x* — ay» + ^ *2r — J\ aft; ^.r* + 3x2+ |-$x«"-3x; |a:«+}x«+^-2x» + ^x*-§x». 16 ANSWERS TO THE 10. 1-f- ia:+^V^'; i^-'-^^' + ^V-i^' + Aa^; ^^a'^ 4 a;*3 + 25 x + 49 + 20 a;i + 28 ici + 70 o^i 11. 9x + 4.x^ + ^xi-{-x-i~12x^ + 2a;i — 6a:J — |£ci5 4-4ccs 5-23 x3i Exercise 22. 1. a'-7 a«^» + 21 a^^2 _ 35 ^4^8 ^ 35^8^4 _ 21 ^2 ^5 ^ 7^^.! — ^>'; a^ + 6 a° ic + 15 a^x^ + 20 a'^c^ + 15 a'ic* + 6 aa:^ + a:«; a^-4:a'c+ 6 a^ c'' - 4: a^ c^ + aU^; a^-16a« + 96a« - 256a» + 256; 16 + 32a + 24.a'' + Sa^ + a^ a^ - 5 a* + 10 a^ _ 10 ^2 _^ 5 a - 1 ; 1 - 5 a + 10 a^ ~10a» + 5a^-a^ ; 16 a*- 96 a^i + 216 a^b^ - 216 ab^ + Slb\ 2. ic2-12ici+ 54ic-108xi+81; a^a:^ -15a4a;« + 90 a^a;' - 270 a'x^ + 405 ax^ - 243a;^«; x' - 15 x' + 90ic« -270a;2 + 405ic-243; Sa^x^-ma''Px''i/-^54:aH'xy'' - 27 ^« 2/«; 16 a* o:^ + 96 a^ ^ a;^ y + 216 a^ V x" y'' 4- 216ab^xy^ ■\- SI b'y\ 3. a« 4- 12 a^ + 60 a" + 160 r^^ + 240 a^ + 192 a + 64; a« - 12 a^ + 60 a' — 160 a^ + 240 w" — 192 a + 64; 16 — 3^ a 4_ I ^2 _ ^a^ ^8 _^ ^ ^4 . ^1^ ^4 _ 3 ^3 ^ + V- <*''*' - 54.ab^ + 81 ^4 . _i^ «4 _^ ^ ^3 ^ _j. ^ ^2 ^2 - 3^ a 6» + ^V ^^ ai« + 10 «» ^ + 45 V ^2 + 120 aH^ + 210 a^ b^ 4- 252 a^ b^ + 210 a* 6« + 120 a^ b' + 45 aH^ + 10aP + b^\ 4. a + 4 4- a-i — 4a^--2a«+4a-i; 4x^+4^4-8:^2^2. l + 3a2_^^4_,_2a4.2a^3; _l_5a2-4a* + 2a + 4a». 5. 16a8i + 64«63; _7 + 20a-16ct2^4a«; 1 + 2 • 3^ • 5J ^ 2 . 2i • 3^, ELEMENTS OF ALGEBRA, 17 Exercise 23. 1. 3a^bj 6ab\ ^abc-^\ f mA. 2. w«; a^ a-H-^c^-''', a?"; 2^'. 3. 5al AJa;-^ ^a-i**; 3a••-^mx'-^ 5. (5c — y)2; (a-c)*; | 6*-i r"' A;«-«. 6. Oa^^c-^ a'"'-"'; 2'm'w"^ Exercise 24. 1. 2a;; -2a*6*c«; -oa*; 1. 2. —3a; -^a^i^c*; .Saft-^c"^; -SOa'^ir*. » a A 5 m~ * a;* V* n i ^ 2 s *• — ^^ «7wy«; — 31 ma: 2/^"-*"'. 5. -6a:"-'/''; - ii(a-^)c"^ ^a-i^^-i* 6. 10 a;l y- i (a;i — - y') A ; | w w x A y. 7. - 4 a* (x - y)^ 2- 1 ; m" n"" (a; - y)^'' (y - «)H. 8. 6a-^6c; -2a'» + "6- + "c-'. 9. -2a-Sil. k 13. -^a^-^-d^-^-c*-"". 11. 2a6*c*rf°x-». 14. Ha"'6''-ic-Ix-». Exercise 25. 1. 1 + 3 ay — 4 aV; 3 m^ /i^ - m 7i - 2 + -?- . mn 2. 1 -J«c- aft r2 + aH*c^ 6a;*— V- a: 4- 40. 3. 4a«+ f a-3 + -; 4a6-6-^-a-^ a 4. 9aftr« — 12aft' — 5c*+ ^aftc. 2 18 ANSWERS TO THE 5. 3mn — rn^n"^ — 3 n -{- 5 th n^ -, x"^ — x^^ ?/'. - 6. 2 a — 3 ^> + 4 c; — J3O- ^^'^ + 2 n". 7. _ 3 ^4^7 _ I ^2^-1 ^ 2 a-i; .9 7i^e^ - 1.2 ?iA. 8. 36f 05 2/" + 10-^-8^; 3mi« + 160m-80 7y^-l7^. 9. ^m-2 _ 2 a-«>-2 4. 3 a!'-^\ m"-2 — m**"^ + ??*" — w" ' * 10. — 3 xy""-'' — 2d^x^ + 4.a'^x if''. 11. _ a»»-i y^ 4. a"* ^> — a'*-^; — f a^ ic'^ + \l a x^. 12. 6 a — ^ ^ — 6'; xi — 3 ars". 13. 3 m5 — f ?j3 + ^ r^if ?ii 14. 4 (x - ^)* - 3 (x - y)2 + 2. 15. 2«'-i + Sx"'?/" — 18a:'"y-'*^'+\ 16. (cc — ?/)''-'* — m«-". 17. (x + yf-' (x — yy-" + (X + yy-'(x — 2/)«-^ 18. 3 7?i — 2 71 — 4 ; «i — a*^ ^>2 + ai Exercise 26. 1. Tx'' + 5xy + 2y^ 2. x^ — x"-^ ?/ + £C z/'-^ ; a^ + or ^ — b'^. 3. vy« _ 2 2/2 + 7/ + 1; 7/5 + 2/* + 2/' + 2/' + .V + 1- 4. JC+22/-.^; a7 + a6^_^^5^2_|_^4^34^3^44^2^5_^^56 4.J7^ 5. 2 a; + 3 ^ ; a^ 6. .25 x2 — 3 X ?/ + 9 7/2. 7. 27x2 + 12x7/+ 6 7/2-1; x^ — x^ 7/ + xif — y^. 8. 1 — 7/ — X + XT/; 2 7/3 — 3 7/2 + 2 7/. 9. x2 — X 7/ + X ;2 + 7/2 + 7/ ;>; + ^2 . ^.3 _|_ ^^ yj 4. /jjj 2/i + 7/i 10. X 7/ + 7/ ^ — X ^ ; X'^" + X^ 2/^ + aJ5 7/t + X? 7/5 + 7/t. 11. 4x2 — 6X7/ — 8 7/2. , ■ • 12. X2 — X7/ + X + 7/2 + 7/+ 1; ^X2-^X + yV- ELEMENTS OF ALGEBRA. 19 13. 6x*y«-4ar^y« + y^ x! + xyi + xiy + yl. 14. aH — ab^j x» + x-\ 15. X- + a; 2/ + y* ; « + "i Oi + 0. 16. a^- ia+ 2; 2*5^*-3«'y+ ixy\ 17. a — aJ ; x* + x*y + x'^y" + x ly' + y^ 18. X* + y-^ + ;g'^ + - xy - X « — y « ; x'^ + .75. 19. x4-2y + z. 22. 2a^ — 3ab + ^b\ 20. |x*-ia;-§. 23. 6aj-^y-i. 21. X* — x'' y + X y* — y** ; x'— x* y + x^ y* — x' y* + xy* — y'. 24. a»-2tt«i'- + 3a*^»^-2a-'^° + ^«; a»+ 2a^i + 2 a6^ + ^»». 25. 2x** — 4x-y" + 22^"; x^'^-x'-y" ^y'". 26. X* — y" + z' ; 3' + 2'. 27. |a^ + ix»y4- ^% x'y^ + ^h^y'i x-i'» + 2y-i". 28. yx" + zx" + r. 29. x-i + y-*; x^ — 2Jxy + y'*. 30. x-J-y-l + sr-i; x» + x''y + xy»+ y'- ^^' X — y Exercise 27. 1. 7/1* + //? n -f J/'* ; a* m* -{- a^ bm^n-\- a^ b^ m^ n^ -{■ ab^mn* -^ b*n*; m* n* + w^ n^ + m^ n^ ■\- mn -\- \. 2. 1 + m n X + m' n* X* + 7w* 7i* x* + w* ;i* x* + m* 7i* x* ; x«»y« 4- xV« + x«y 2* 4- x\//2* + xV^* + a;«yz» 4- x*2% 1 + ahX'\-a''b'^2? + a''Vv+ a*6*x* + a'*6«x'» + rrV/«x«. 3. a«+rt»^ + ft*; x" + x'y' + y"; x" + x^V + x»y< + x«y« 4. a" + a»i« + a*i"+ aH^.+ i^; x""H- x"- 7/- + .r'*"?/"'- + 2*-x*- + 2*-x-"' + x*-. 20 ANSWERS TO THE 5. 16 a^ + 12 a^n^ +9 n^', 4:x^y^*'+ I. 6. a'^x^P + a*b^x^pf"' + aH^x'^y^"' + b^y^'"', 16x^+2Ax^y^ + 36x'y' + 54.x'2/ + 81^1 7. x-s + ic-K?/-^ + x-^^y-i + y-i\ xi + ic-^/^ + »^^y + ^V^ + x^iy^ + y}', a^ x?+ ai b-^x^ y^s + a^ b-^xyi+ai bx^yi + b^y^. Exercise 28. 1. 12ba^x^— l^a^mnx'^ + ^5 am^ 71^ x— 21 m^ 71^', x^-x^b^ + x'b^ — b^; x'' — x^ + x^ — x^ + x^— x^ + X — 1. 2. £c5 - 2/?; 64 a;^ - 160 x^ + 400ic - 1000; x^"" - 7/\ - 1^5 2/^ ; a^-aH + a' h' -- aH'' ■\- cv' b^ - a'' b'> + «' ^' - a^ ^7 + «, ^8 _ ^9^ 4. 243 ai<> - 162 a^ b^ + 108 a« b^-12 a^Z/^ 48 a2^»i2_ 32Z;i5; ax" — 5'-^ 2/2"*. 5. a-^ — a-^x-^ + a- 5 ic- « — X- ^ ; a^ x~ ^ — a^ ^s x-%~i -\- ab^ X- -^ y-^ — b^ y- 1 6. icl« — £c"2/T2'" + xi'*?/i'" — x^"2/i'"' + xi"?/3'" — ?/?5"'; Exercise 29. + ^-22/-2_.^-iy-3+ y-4. 2o6x'-192x'y + lUx'y' -10Sx7f + Sly\ 2. 64xi«- 96x1^7/2+ 144xi2y4_2i6x« 2/'+ 324x«2/'-486xy^ + 729//2; Slx^^-5ix^7f + 36x'y'-24.x^y'-{-16y\ 3. Xl2n _^iou ySm _|_ ^8«y6m_ ^6« ^^9m _j_ ^4« ^12m _ ^2n ^15m _|_ yl8 m . 7^18 a_jA6a ^^5 n _|_ /^12 « ^^^10 ri _ ;^9 a ^^^15 n ^ ^6 a^,^20 » - A;8« ^25" + m»^" ; a^^ - a^ ^' + a^ b^ — aH^ + a^ b^ - aH^ + aH^ -aH' + aH^ - ab^ + b^"". 4. 7^1 ^1 _ mi 71^ x^ 2/1 + m§ ?i? £C^ y^ — 7?^^ ti^ x^ ?/5 + x^ y^ ; x^ — x^y^-^xy^-x'^y^ + y^', x-'^-x'^t/-^ + x-^ y-"" - x-iij-^ + x-Uj-^ - x-'^y-'' + y-^. ELEMENTS OF ALGEBRA. 21 5. af X* — a» 6A xi y» + a^ b^i x^ y^ — a^ b^ xl if + a? 6i? x y> 6. a'^(iH'-\-b^; a;*-a:Hl; a:«-a;* + l; ai8-a«6»4-<^". 7. a;»-x«/ + «*y*-^'/ + /; a;''-xy + y°; 16-4x^0;*. 8. 16a;*-36a;«/ + 81y^ a;*-^ «« + tV ^'- b^ a;'+ ^i^; 9. ,,12 _ ,,« // _|_ /,ii 5 «i« _ ai4 ^2 4_ ai-2 ^4_ ,jio ^e _|. ^8 6« - a« ^»^« 4- '*' b'^ - a^ b'* -h b'' ; ,V a^' - 3^ ^' f + iV y'- 10. >(•'* — a» ' ^^•- + b''* ; a«-^ — rt^s ^,4 ^ a'" ^»8 — a*' ^" + a^« ^^^ - a" &» + «« 6'^* - «* />-^8 + 6«-^ ; 81 x* - 9 a!^ + 1. 11. .r"' - ./•'» / + x""* i/'-' - x" i/^ + ^12 2/24 _ a.6 2^80 ^ yj6 . ^m ^12 _ an» + «• ^' - <^' b^ + ^''. 12. .r3« - x" //" + y»« ; x« - x^^/ + x«« //^^ _ ^so ^is ^ ^24 ^84 - x" 1/^ + x^- //« - x« //« 4- y/*8 ; yes ; a« _ «« ^4 _^ ^,8 . a4 ^« _ ,^2 /,8 ^9 ,^12 ^ ,^^18 ^^24 . ^12^^20 _ ^9 ^16 ^,^6 ^2 + a« 6»« m" /i^ - «» 6« wi" 7i« + m" n*. 13. 2 + ./, 4-2a + a2; a» + i', </«-^»«; 2 - x, 4 + 2x+x'^; x«+9, x«-9; a*^ — 6*, a^« + ^/6^;<+ ^^8; 9««^-4^»^ 9«6_4^4. a2_|_25, aa-25; a»-J», ««+a«i« + i«. 14. x*-y», x^+x^V+^V+aJ^iy^+.v"; ^w+x, 7«*-7//«x + tTi'^x^—mx*-\-x*', x« + .y«, x^ — i/; x* + 1, x« — 1 : a-« + ^-*, a-« — &-•; a*x*' + b*y*', a»x»' — (^« ?/''; a; + 2,x*-2x« + 4x«-8x+16; 4a« + 9, 4a''-*9. 15. 2a — 6, 16a* + 8rtV; + 4an«4- 2«V>8 4-i*; 9a< — 4/>, 9a* 4- 4ft ; 1 - y, 1 + y + y* + f-\- y* + ?/ 4- ?/; oa; + 10, a«x«-10ax4. 100; a^x'^-l, a^x'' 4 1; a + 7WX. rf^ — r/'/nx 4- a^ m* x!^ -- am*x^ + m*x*j X y — 9 ^, X y + 9 a. 22 ANSWERS TO THE 16. 2 a^ _- 3 h% 16 a' + 24 a^b^ + 36 aH^ + 54 aH^ + 81 b^""; + b^""; a^x^" + b^i/^"\ a^x^"" — a^ b'^x^^'y^''' + b^y^""-, cxP + b y% c^ x^P — c^ b x^^ y"" + c^ b^ x^^ y^ » — c^ b^ x^^y^ " + etc. * 17. a;-2" + 2^-2«, x-'*" — a;-2"2/-2"+2/~^"; 2a;y+9, ^x^y"^ — 36 a; 2/ + 81 ; a^ f» — b^ /", a^ 2/«^ + ^;« 2/«« ; c' x'' + ^>* ^Z"", c* x^^ — b^y^""', XT — 36, a;-^ + 36 ; a^" — ^2«^ 18. 2 a;8 + 3 3/'^ 64 a;i« - 96 cc^^ ^/^ + 144 cc^^^/^ - 216 x^ y^ + 324 x^y"" - 486 a^^ y^"" + 729 ?/" ; 16 a;« + 9 2/^, 16 ic« — 92/^; a'"6" + ic^/, a^"'¥'' — d^'''h^''x''y'-^a^'^b'^''x^'y'^' — or b" x^'' y^' 4- x^^y*' ; l + 2x\\—2 x^ + 4 ic^ — 8 ic« + 16 a;« — 32 a;i<^ + 64 x^^; a"*" — 6«", a"*" + ^«»j x-^y-^ + Ij x-^y-i — 1. 19. ic^r. ^1 _ ^^1, y-j^ ^2„ ^-1 _l_ ^^^-i. a-^jc-i+ 1, a-2^-2 — a-^ic-i + a-^x-^ — a-2ic-4 + l; fa^ici^H- |^'~^2/"^ — i 2/-^ tV ^-^" + ?^T ^-"2/-"' + T^o x~l^y-^^ 20. ^'s ci« + .3 £ci ?r ^ ^^ ^i'' — 3 ^> cf « ic^ ?/- ^ + .09 b^ c?« a;? 2/~^ -.027 ^>i d«a;? ?r ^+. 0081 a^t 2^1; ^16a^ic-f " + .090!-^"', 16a5a;-l«-.09a;-J'"; 2-t«aTV-3^ i-^,2-§"«V<. + 3^^»-i Exercise 30. 1. 3 a^h-^ — ^^- a + ^V" ^; ^~^ — ^^y~^ + .V"^- 3. l — 2a — 2aM',xl-\-xh y\ J^ x^Tjh ■\- ?/l. 4. {a — b — c)"^ — (a — b — c)-2'« — (a — b — c)"". '5. 0* + y + ,*; ; a?^ _ 2 a' ?/ + t/^. ELEMENTS OF ALGEBRA. 28 6. x* — 2x^f/z-{- 4y-^2; ar^ -f 3xy + '^^xz + 3/ + z\ 7. a;-2y + a;ijri+y'. 8. 2 X*" — 4 x"y" + 2 y"^». 9. a;" y — af " * i/^\ 10. a'^- - 2 a^b" + 6=^"; a^' — 1 ~ a--^'. 11. 3a*'' + ^ — 4a«'' + 2a=^«-^ — a«-2. 12. 2al — 3a->*« — a-i\ 13. Sx^ -42f-^ + ox'-'^ — af-\ 14. 2 m'-^ + 3 m'-^ — 4 ?/t'-». 15. Saf — 4.af-^ -\- Saf-^ — af-^ 16. a"'"— ' — ^("-^>"'. 17. 2a; + 1, 4a;''^ — 2a;+ 1; 4 + 9 a^ 4 — 9 w^; 4 ^ - 26, 16a''« + 8a6 + 4 6'^; a + 10, «2-l()« + 100; a^ — S, a* -\- S] m — n, m* -\- m^ n + w,^ n^ -\- m n^ + n^ ; l--2y, 1 4.2y + 4/; aJ-1, a«6«+ a«6^ + «*Z»^ + ««6»+ aH^H- a^> + 1. 18. a;=^+a:-^a;2— a:-2; WaX^'x^^— ^^^ul'^h^x^^ -\- ^^aV'h^'^x^^ - iT.?«^''*"a:'''+2^?Z>^'; a:^" + y-*"', a:^" — a;*«y-^"' + y-«-; 05' - y^ x^« + a;^^ 5^ + a:^* y" + a:'y" + y«> ; 3.6m _ 2.8m y8H_,.y«« 19. 2a« — 3y-», 4 .r< -f 6 a^y* + 9y-«; 4 a* — 3 7i-», 16a» + 12 a* n-« + 9 w-«; ti «*" ar^ + I ^*, 3 aA" + 2, 81o!- — 54^A'' + SGrtJ" — 24aA" + 16; cx« — ay"', c^a;*" 4- ac^x^^y"' + a^c'^x^y^''' + ««^«"y»"' + a*y"; J a^- + .04 6J% I a^"- .04 ^J"; 8" a*" + 9", (64)" a^" - (72)- a*" + (27)*. Exercise 31. 1. ± 5a;y^; ~2a26a;«; -5«i''; ±3a*i^ 2. _7aW>-«; ±|ar^y*«*; -a^V; i-'^^ST'- 3. a:*; ± 11 a;«.y; ±5aft; ±2^-26^ 4. — 3 a" ^.-*; - 4 7/1 n^ a-«: 7»^ w«. 24 ANSWERS TO THE 5. -la'-y-^', 2a3x-2; ±^0"})^. 6. la^hc^'d-^', ±^0,-^1)"', 2; - 2 a». 8. £c'"^; 2a2 6*a;«; iOir^^'^+^j _2cc"-2/+s^ 10. ± 2x''y«"^'^; — fm-^Ti-i; a6^c-\ 11. \\ a bi c- i, or — \% a b^ c-i. ^ i_ 13 m — l 12. V84; 5''xf; 3»aH"; a-i; xy, x » /; icy-^; a^b^^c^\ {x + y) (x - yy. 13. (a h^ c")"; «'» {x — y") ; a'^ x^p ; (a? + ^)2«. Exercise 32. 1. y — 1; 3 a^ _ 2 a — 1. 10. 1 — a -^ a'^ — a^ -\- a\ 2. 2a3-3a26— 5a62_^76^ '11. Sm-zz + '^^^r + y. 3. x^-6x^ + 12x—S. 12. x^-3x^y + 3xy^-y^ 4. a2+2a + 2. 13. 5a;2_ 3 ^^ _^ 4^2 5. 3 + 5 X — 2 x2 + x8. 14. x^-3x^ + 4:X — 5. 6. ab — 2ac + 3bG. 15. 2 — 4 ai + 3 M. 7. 7 a^ _ 2 a - |. 16. ^x^—xy-\-^y^; x^-3x^-2. 3. 2x + 3y — r)a. 17. 5 a:? — 3 £c* + 4. 9. m^~3am'' + 3a''m — a^ 18. ip^ _ 2^2 _ 3^-1^ Exercise 33. 1. 182; 6.42; H; ^^; .315; 1.082. 2. .5555; 75416; 30709. 3. .2846; .9486; .0316; .3794; .5000; .0169; 1.8034; 4.5728. ELEMENTS OF ALGEBRA. 25 Exercise 34. 1. a^^x—1; x^—ax—a^. 7. a — b — 2c. 2. 2x' + Aax-3a\ 8. 1 — x -\- x^ — x\ 3. x^-2x-{-l. 9. 2x-^-3a;y + 5y^. 4. 3a'»-2ci6~^. 10. a + 26-c. 5. l'x^-l-x-3. 11. x"^ + a;y-2/. 6. .3x^ — xi-6. 12. 2y- — 3x2/ + 4ar2. Exercise 35. 1. 42; 32.4; .625. 2. ^^, or .0425; .0534. 3. .861; .430; 2.017; .669; .200; .873; i ^i-i, or .637. Exercise 36. 1. anbi'^crl; 200 a;» (* — .V") (« + JTY (^ — y + «")*• 2. — 32a\/2axy. 3. Sx^-2 -\-x-l 4. 2xi'' — 4 + 3a!r-i". 5. 4 a;"' 4- 2 aj*" — x'*". 6. a;iy-J — 2 4-a;-'y*. 7. aa:»-2 6a!« + 3c. 8. i x« - 2 X + i a. 9. rt" Ta;'"; a4±x*. 10. 2^^ + 4a;y- 3a;«. 11. x^"" — 2a?^y" + 4y*". 26. 3 a* — 2 a + 1. 12. X-* — 2ar> + l. 27. x-^ + af — aj. 13. 2x--iy«. 28. a- 2. 14. 3a-J — i + 2a-J«. 29. (a + ft)^"* x + 2 a"r. 15. 1—3 a. 30. x" + x"-* + x«-*. 16. X— y. 31. 2 — a»"-^ 17. 2a + 1. 18. ±12; ±8. 19. a 4- 1. 20. a^-ab-{-b\ 21. 517. 22. 384. 23. a^ — 3 a + 5. 24. X* — (m -I- w) 25. 5 a -2^ + 3 26 ANSWERS TO THE Exercise 37. 1. 4 c. 18. a + 2b — IS G + 7Sd. 2. a — b + c. 19. — 6 a. 20. 0. 3. —Sx^ -Sx. 21. 4 a2 _^ 4 ^2 _^ 4 c2 + 4 d\ 4. —11 X - 2 y, 22. 0. 5. 3 a - 8 Z> - 2 c. 23. 12 ^8. 6. — 35 « + 30 ^> — 30 c. 24. — 3 a 7i 2/. 7. 4 ^ - • 16 ^ — 2 c. 25. 2 ny'^. 8. X h 2 y. 26. Saxh — 3m + 6 n. 9. 3 b. 27. 0. 10. 2xy-- 'J y — z. 28. 3m'^n^ ■\-2 m^ n^ — n\ IX. ^^^a + S. 29. ^jy^ + ^xy. 12. a -- J-^ b -Jr Y- c- 30- S x"^ — 8 y\ 13. 3 a 4- 4 :?. 31. a (^ + c) + 5 c. 14. 7x'-hy. 32. (a + /*) — 9. 15. 210' Z>- 222 a +84. 33. (a; + ?/) + ;^. 16. ^- xh. 34. (:*; + ?/) — z. ^^ ^^-^CL' 35. (a + ^,)2_(a+&) + l. Exercise 38. 1 «4 _ j-^ ^3 _^ 5 ^2 _ 2] ; ^5 _ [6 w2 — 3 m8 - 3]. 2. 3:r-[22/-5^ + 4?^]; »«/>«- [2 a^ ^^ _^ a ^»« - Z.^]. 3. A:X-\-3ax^—\Qx^ + Bcy~y'\\ x^ — y^—[z^—ab — 3acK (3 or - 2 y) + (- {4 7Z - 5 ^}) ; (a^ b^ - 2 a^ b') ^(^_{ab^-b'}). (4:x + 3ax^) -(6x^-{y-5cy}y, 5. - [3ay - 2 ab-] - [5 b x - 4. b z^ - \2 c d - 3], -[3a2/-2f^^'-4Z*,t] - [5^>ic + 2^cZ + 3]. ELEMENTS OF ALGEBRA. 27 6. - [- a + 26] - [rf - c«] - [1 - ;.] _ [aj + 2y] — [/i — 2 //t] — I4:abc — jj], — [2 6 — a — c z} — ld-{-l-z] - [a; + 2y-2 w]- [n + ^abc-pl 7. - pxy-2x] - [5a:«/-4x^y-^] - [xyz-x'fj, — [3 j:^ — 2 u; — 4 x-y-] — [5 ic^'y- + xy 2 — x*y^]. 3. _[_x* + 4«»]-[3a=^-3a^]-[l-a],-[4a»-3a*-x'^] -Pa'^-a+lj; - [2 w-4j»] - [37i + l]-[5x + 6y], — [2 m-}- 371 — 4/)] — [5x-f 1 -h6y]. 9. — [rtc — a/i] — [c'x- ai] — [ax + ayj — [3aic — 3x^2], — [« 6*— a ?t — a 6] — [c x-\-a x-\-abc\ — [ay+ 2a6c — Zxyz\. 10. (2a6-3«y-|-46«)-(56x-[-2c'(i-3]). (a-2^* + cz) _|.(2_</_l)-|-(2 7;i — X — 2y)-(7i — [— 4a6c-j9]). (2 X — 3 X y 4- 4 X- y^) — (5 x^ y- — [x* y' — xy .^]). (x»-|- 3 a* - 4a«) - (3^2 _[«_!]); (4 j9- 2m -3/1) — (5x— [— 1— 6y]). {an-\-ab—ac) — {cx-\-ax-\-\ay) ~{h^y~ [jixyz — 'dab c]). 11. m'-hem*-^?*-!- 12?^i7i- + 8 7i» — 3m2x — 12mwx-12 7i'*x H- 3 m x^ -I- G 71 x^ — x^ Exercise 39. 1. 9; 1. 3. 4; 2. 5. 1. 2. -A; i- 4. 21; 25. Exercise 40. 6. 1. 1. 16; 9. 4. ¥; 1- 7. 4; ]^. 10. 3. 2. 12; 5. 5. 0. 8. 1%. 11. 2;-6§. 3. 1; 60. 6. Exercise 41. 12. 9 1. -1.7. 4. 66§; 20. 6. 2. 8. 8; 270. 2 3 5. 5. 7. 5; 9. 9. -t;i- 3. 11. 10. csj. 28 ANSWERS TO THE 2. 3. 4. 5. 7. 8. 9. 10. 12. 13. 14. 15. 17. 18. 19. 20. 22. 23. 41. 42. 43. 44. 45. 46. 47. 48. Exercise 42. 12, 17. 24. 2 at 65 cts., 22 at 35 cts. 17, 31. 25. 25 lbs. $20, $30, $40, $50, $60. 26. A, 60 ; B, 10. Father, 48; son, 12. 27. A, 72; B, 24. 28. 7 years. 29. 40 miles. 30. A, 28 ; B, 14. 31. 103 gallons. 32. 3.7. 33. 20, 21, 22. 34. 24, 25. 35. 28, 29. 36. 100. 37. 200. 38. Watch, $117; chain, $68. 39. Linen, $0.32^; silk, $1.95. 1. A, 25; B, 20. A, 65; B, 40. A, 25; B, 5. 8, 9, 10. 2,5. 5. $50, Father, 36 ; son, 12. 100, 200. $23.40. 162. First kind, 21 ; second, 42. 40. 11. 77 at 13 cts., 11 at 11 cts. 21 three-cent pieces, 18 five-cent pieces. Son, $1.04; father, $1.41. 76 ten-cent pieces, 19 twenty-five-cent pieces. $70, $52. 25 dollar pieces, 10 twenty-five-cent pieces, 20 ten-cent pieces. Florins, 53; shillings, 71. 10 shillings, 20 half-crowns, 5 crowns. 40 guineas, 52 half-crowns. ELEMENTS OF ALGEBRA. 29 49. 4 children, 20 women, 60 men. 51. Oats, 20; corn, 40; rye, 120; barley, 480. 52. $1225 at 7%, $1365 at 8%. 53. f 180 at 15%, $150 at 8%. 50. 30, 10, 223. 55. Saltpetre, 6; sulphur, 9; charcoal, 6. 54. $5070. 56. 51 women, 65 men, 150 children. 57. $3.75. 58. A, 47f miles ; B, 37| miles. 60. 10^ cents. 59. $313^ at 15%, $16§ at 8%. 62. 7. 61. Horse, $375; carriage, $300; harness, $75. Exercise 43. 1. 2x2 X<iX ax (I Xbxbxbxx; 2x^XxXXXxXyXy', [\X^XaXbxbxbxcXc\ 2 X 2 X 5 X « X ^ X <; X r Xc; 5x7xa;XiFXa;X2/XyX«X«X«X2X«X«; ^XlXaXbxbXxXX', 3x3x2x 2xaXbxbXxXxXx. 2. ^aHxA^a'b', 3x«y3 X 3xV; "d ab'^x'^ y^ X*d ab'^x'"' ^', 13a4-6i X 13ai"^>i. 3. 4aU*; 3alMx; 8ai6ic2; 5a-ift-^x»y». 4. x\'X^x\x^xix^x^\ m'" wJ^.m?", ?Aii''.mi''?/ii''.ml''; a;4 . xi . x\ a;i • jc* . a;i . jc» ; xl xl- xi, jc* • a:* • xJ • xK Exercise 44. 1. n{77i + 1); ab(4:a + ftc + 3); 3a*(a- 4). 2. xia-b-\- c)', x*y^ (39 y» + 57 x""). 3. x«(5 a; 4- 7); 12bxy^ {6bx - 7 i^ - S axy). ^ 2 aaf y z {462 at^-^ — 5S9 s:r-^ + 616ay2). 5. 4 a 6 (a - 1 5 6* + 5 c + 2 a 6» a;* + 4 y - 9 a« c ari) . 6. a;iy(2 — a6a;l + cajly«); 6a;J(a; + 2a;i — 3). 7. iac»(i<r-l + a-n-4a4&-lcl); a"a:"(a*-- a"x'' + x2"). 8. a''^"c*(ft"<r'"4- a'-ft*" — a^^c"). 30 ANSWERS TO THE Exercise 45. 1. (x + 8) (x + 11); (x - 3) (a; - 4); (a^ - 8) {a* - 12). 2. {x + S)(x + 27)', {bc-n)(bc-lS). 3. (aH^ + 12) (a^ P + 2o) ; (« + 11 ^) (a - 6 i). 4. (6^^-8)(a^' + 3); (^^ + 4) (t/^HH); (aH5)(a«+12). 5. (ab-]-2c){ab—5c)', (a'' +10) (a^— 12); (n + .5) (71 + .3). 6. (^'^+762>)(a2-8^^^); (x+5)(x-U); (x''-\-25a'"){x''-12a'). 7. (i:c-4)(aj-ll); (m + i)(m + |); (oj + 2)(x- 13). 8. (a& + 5)(a& + 26); (a-5^>:z;) (a- 15^»ir) ; (^/HSa;^) {i/-9x^)', {l + i)x) (l + 7x); (m-7a){7ri-Sa). 9. (a + 9x1/) (a - 21 xy) ; {x + ij + 4.) (x + y + 1), 10. (1 -5a&)(l-8a6); (a - ^ + 2) (t^ - ^ - 1) 11. (x-y+2){x-y-b); {x + 21) {x + 21)', {x'-12) {x^-ll). 12. [(a + Z^)2 + 1] [(a + ^)2 + 8J ; {x^^'-b){x''--m). 13. (a + 3 ^>2 c) (a - 13 b^ c) ; (ic« + a) (x« - b). 14. (ic + 5 2/) (a; - 14 2/); {X + 1) (a; - I); (x^" - 20) (cc2«-23). 15. (aj + 1) (oj - 1) ; (x' + 21 C.2) (^2 _ 22 ^2), 16. (x 7/ + 11) (xy — 14) ; (a" a;^'" + 11 if) (»" :z;-'" -f- 3 ?/«). 17. (a; ?/ — 11 a'^ 6'*) (a; y — 17 a''^*") ; (a; - ^) (cc - 1). 18. (a;2»2/'"+17a'"6'")(£c2«^2«_^3^m^.„-). |-(j^_^^)3m_|_7^4nj [(aJ + ?/)'"•- 14 a^"]; {n'+ .11) (^i^-.l). 19. (o^ + f ) (a^ - I) ; (:r + 2.1) (:r - .1); (a' + 1) (a^ + i). 21. (2 cc - 2) (2 X - 3) ; (3 a: - 3) (3 a: - 6) ; (2 :z; + 6 a) {2x + 2 a). 22. (Sa + 4.b){3a + 6b); {4.x-2a) (ix- 3 a). 23. (5 x^"" + J a") (5 x'^'" - i «") ; [6 (a - b)^" + 13 (a — b)'] [6{a-b)^" - 11 (a-b)^]. ELEMENTS OF ALGEBRA. 31 Exercise 46. 1. (4x+l)(x + 3); (2y+l)(2y-3); {3a^-\-x') (ia^-x^. 2. (3-a:)(H-4-c); (4 a; + 3y) (2a; - 7 y); (3aa:-l) (2aa:-f 1). 3. (?/i'-3)(8w» + 9); (3a- ll)(r>a- 1); (2a + 3Z>) (3 a — 6) ; (2 m — n) (m - G n) ; (3 a; + 4) (a: + 1). 4. (8 + 9a)(3-8a); (a:+ 15) (15a:-l) ; (44-3a;)(l-2a;). 5. (3x~2y)(2a:-5//); (4x-3y)(2x+5y); (a:-5)(15x-2); (12 x - 7) (2 a; + 3) ; (a + 3) (11 a + 1). 6. (3-5x)(6~x); (3x + y)C2x-3y)', (l + 7x)(5~3x). 7. (8x + i^) (3x - 4y) ; (2 x2« + 7 y*") (3 x'^" - y^). 8. (x4-y + 2w + 2/i)(2x + 2y + 7?i + 7i); (x + 4)(2x— 7). 9. {x + y^3a-3b)(2x + 2y-a-b)', (x+|)(Ya:-l). 10. [(x-y)»--2x"'yi«][ll(x-y)»''-x-y5"]; (3a+l)(9a-l). 11. [2 a" + 3 (x — y)'""] [2 a" + 7 (x — y)""*]. Exercise 47. 1. ^m2H-7i2)2; (;,i_|.,i)2. (^ a'' - W b cY', a^tt - 2)^ 2. (7 w» - 10 7i'^)2; (9 x^ y - 7 a')^ 3. (7ii» - 7*)^ (1 - T) wi7i)»; x^ (x + 1)'*. 4. (« + /> + 8)2; (7/1 + 0)^ 5. x*(2«»-5xy)2; (19 a^»c- 2 rf 77i7z)«; (11 7/i7i2_ IOje?)^ 6. (15x2~y2)2; {2a"'-b'''y. 7. n«(7 7w + 3^)^ (:^+ i)^ 8- (s«*+ i*')'; c(aM-3fc»)«. 9. {3x-},yy', {m--7i-\-iy. 10. (a2-a + 3)*; (2x + 2y-f 1)^. 11. (ai - il)«; (;/ii - 1)^; mn (ml - niy. 12. (xJ + yi)«; {m ni - a)«; (2 xJ + 3 /t)'. 13. i(a^br-r,rY- {^,xl--^)\ 32 ANSWERS TO THE Exercise 48. 1. (l-7x)(l + 7x + ^Qx')', {2x-9f)(4.x''+36xf + Sly'); (6 x''-a)(86x' + 6ax + a"). 2. {xy — ah) {x^y^ ^r cuhx^y"^ ^- a^h^x^y" -\- ahx^if \x''y')', \x _ 1) (a;6 + a;5 + cc* + a!^ + a;2 + ^ + 1) ; (3 a - Z>)(81 a* + 21 aH + 9 a?h'' + 3 a ^>« + h')-, (a b"" — m) {a" b^ + aPm-{- in''). 3. (6a-7)(36a2 + 42a + 49); 8x{l-3x){l + 8x + 9x''). 4. (a8 - 4 ^2) (^12 + 4 aH^ _^ 16 a« ^>* + 64 a^^*^ + 256 h^) ; (9 x-12y) (81 c«2 + 108 ic^ + 144 ^2) ; (^^-i _ y-i) (a;-* + x-^y-^ + ^-^^/-^ + x-^y-"" + y"'). 5. 5ir2(3a;-4)(9a;2 + 12a; + 16); 2 a6 (a - 2) (a^ + 2 «» + 4 a2 + 8 a + 16); (ar-i — 2/~^) (ic-? + x-^ y-^ + 2/~^). 6. (ab-xy){a^b^-{-etG.); (4.a^-5b) (16 a* -{-20 aH + 25 P); (iC" — 2/"^) (^2« _j_ j;c«ym _j_ y2my * Exercise 49. 1. (2 a 4- 1) (32 a^ - 16 a» + 8 a^ - 4 a + 1) ; (l + ic)(l-a; + a;2-ccHa;^); (ic'+2/') (^'-ic^2/'+2/'); (^' + 2/') («' - :e'z/' + ^^^ 2/' — x^ y^ + y^). 2. (a 4- 2) («« _ 2 a^ + 4 a^ - 8 «» + 16 a^ - 32 a + 64) ; (a;2+ Oy) (cc*— 9x2^^+81 ?/') ; (4.t2+2/2) (16a;^-4a;2^2_j_^^). 3. (ab -\- x^y^) (a^b' — a^b^xy -{- aH'^x'^y'^ — abx'^y^ -\-x^y^); (x^ + 4 y^) {x' - 4 x^y^ + 16 2/4^). (10 cc + 11 2/) (100 x^-110xy-{- 121 1/'). 4. (x^ + y^) (£ci2 - cc^ 2/' + 2/''), ^i' (^' + 2/') (^' - a;' ?/" + ?/') (a;i2 _ x^ if + 2/1^); 5 x!" (3 ic + 4) (9 tc^ _ 12 a: + 16) ; (a;8 _|_ ^8) (^.^16 _ ^8 ^^^8 _^ ^16^^ ELEMENTS OF ALGEBRA. 33 5. {X-' -f y-') (X-* - x-^y-' + x-2y-2 _ etc.); {x^-^y'-'Xx'^^-xh/^-y^) ; (xy-^ab){x*y'-abxY-\-etc.) ; (a' -I- n (a" - a' i" + 6»«). 6. (a" 4- *") (a" - a" 4" + i**) ; (1 + x') (1 - X* 4- «») ; (af> + y2«) (x2» - x»y2"» + 2^"*); . (ar-» + y-*) (x-i - x-iy-^ + yl). 7. (a*" + 6»"') (a^" - a*" 6»'» + ^"'*); (2a^»c + 3 a;) (16a*b*c*—2-iaH'^c''x + S6a'b''c^x^—54:abcx^-\-Slx*)', (4a + i-^) (2o6a* - 64a»*^ + 16aH*-4:aH^ + 6«). 8. (4x'^4-9a«)(16a;*-36a«a;2 + 81a*); a «' + J *') (^ «* - ^'ff «'^' + tV ^*); (a« + 6 c) (a* - 4 aHc + 11 ^^^c^). Exercise 50. 1. (ax + by) {ax - 5y); (4x + 3y)(4:X - 3y); (5aa; + 7 by^) {5ax-7by^). 2. (ar^ + 5^)(x+y)(a:-y); («^ + 9y2)(x + 3y)(aj - 3y); (x* + y*)(ar^ 4- y^)(x + y)(x- y)-, {x' + y)(a;* + f) (x'^y){x^-y). 3. (aH^ + 9a;V)(«'* + 3a:yi) {aib-Zxyi)-, (l-\-10a*b'c){l-10aH^c)', (4^8+ 3 6«) (4 a* -3^"). 4. (3a'' + 2ar»-)(3a"~2x^-); (^ a + ^ 6) (^ a - ^ 6) ; (a;J + y*)(a:i-yi). 5. (a;-«+y')(a;-'4-y)(x-*-y); (a4-^+c4-<^)(a+^-c-rf); (x— y + a) (X — y — a)- 6. (a4-a:—y)(a—a:+y); {ab-\-xy-{-l)(ab-\-xy—l)', 4ab. 7. (a 4- Z^ + 2) (a - ^») ; (a 4. 6) (a - * 4- 2) ; 503000. a 47a;(aj4-2y); 2805000. 9. 12 (a; - 1) (2 a: 4- 1) ; 1908 X 1370. 3 34 ANSWERS TO THE 10. (a^« + 1) (a^n + 1) (a« + 1) (a» - 1) ; xy{Sx + y) (9^2 — Sxy + f) {Sx — y) {^ x' + 3£cy + t/^) ; h {a" + h^) (a + b)(a-b)', {ah + bh) (ah - bh). 11. 6t (a + 4 a; — 6); (2 aj-^ + 3 2/"^) (2 a;"^ — 3 5/-1) (4 a;-- — 6 a;-*j^-i + 9 y-^) (4 x-'^ + 6 a-^y-^ + 9 y-^). 12. 150000; 2{ah-\-2hx){ah-2hx)\ (5a4"+3i^>'«)(5a5"-3i^>'»); (a: + y) (a; - 2/) (x^ — a; y + y^) {x'^ j^xy + y^. Exercise 51. 1. (a -^ b + c) (a — b — c)-^ (a -\- b — y) {a — b — y)-, (^a-b + 2) {b-a + 2). 2. {5x-\-b-{-3c){5x—b — 'dc)\ (a-\-x + y-\-z){a-\-x—y—z). 3. (2a;-3y+9)(2a;-3y-9); (a; + 3) (a; + 4)(a;2-7a;-12); (2 a: - 1) (a: - 1) (2 a;2 + 3 a: - 1). 4. (4a;2+a;-^)(4a52-a;+i); (3a+:r+42/-l) (3«-a;-42/-l). 5. (a; + 5/ — m + w) (a; — y — m — 7i) ; (a^ + ^^ + c^ 4- ^^) (^2 _^ ^2 _ ^2 _ ^2^ . 4 ^2« (^n ^ J«>) (^« _ J«)^ 6. (;s + 2a!— 32/)(^— 2a; + 3y); (2a + l— 2a;) (2a-l + 2a;). 7. (x + 3^ — ^) (a; — 2/ + «) (ic + .y + ^) (a; — y — «) ; 8. (c + (^-3a + 2a;)(2a;-3«— c— c^); {2x—Sy-{-4.z-\-bd) {2x-Sy-4.z^5d); {b + c + 2x) (2x-b -c). 9. (5a«+4a2+a;2-3)(a;2+4a2_5a84-3); (y-^bb + Sbx-^l) (3,_5J_3^a;+l) ; a2„^^«.|_2)(a"-2) (a2«_6)(a2' +2). 10. (3 a 4- * + ^"^ — y"*) (a;" — y*" — 3 a — J). 11. («» + a^^" + y - 3 ;s) (a^ + xS" + 3 ;s; — 2 ?/2m). 12. (2aj + 3?/ — 67?. — 4^) (2a; — 3y + 6 71 — 4;?). 13. (a'' — 6« + c"* + ^^m) (^« _ j« _ c"* — A;2'»). 14. (2a + 3a; + 4?/ — 8«) (2a + 3a; — 4?/ + 8«). 15. (a + lb-3c)(a + ^b + 3c); (a^+ a-b"-- 3) {a''--a-b''-\-3). ELEMENTS OF ALGEBRA. 36 Exercise 52. 1. (3a^ -^ 3 ab + 2b^) (3a^ - Sab + 2b^', (a^ + 3a-{-9) (a'-Sa + d); (-ix^ + 2 xy + y^){4x*-2xy + y^). 2. (x' + xy + y^(x^-xy + y^)(x*-x^y^-\-y*); (9 «' + 10 a a: + 4 x^) (9 a^ - 10 a'x^ + 4 x^) ; (m* -\- m7i -{■ n^) (m* — mn + n^). 3. (2 X* 4- 2xy + Sy") (2 x^ ~ 2 xy + Sy^) ; (a* + aH + b') (a* - a^6 + 6*); (9 a" ^ 6 a + 4) (9 a« - 6 a + 4). 4. (5a» + 7aUi + 46»)(5a»-7ai6a4-4^»«); (aj + ^iyi+y) (x-iciyi + y) ; («» + a;3 yJ + y^) (a:» - xi yi + y»). 5. (4a*+4a^^>i + 36»)(4a*-4a'*^>i + 3i»); (3tt^+2a62+76*) (3a« - 2a62 + 7^*); (^ + jo* + 1) (/> - p^ + 1) (;?«+;? 4- 1) ip" -P + 1) (i?* -i?* + 1). 6. (7aH4a6 + 96^)(7a'^-4a* + 96*); (3a;H3a;yH5y*) (3x^-3xy^ + 5y0. 7. (m** + m- + 1) (ttj'"* - m" + 1) ; (x^^ + 4x» + 16) (x«"-4x"+ 16). 8. (a 4- a* ** — ft) (a — a^b\ — b)\ (a«" + 2 a-6'" - ft^"*) (a*" — 2 a" 6™ — i^'"); (5 m* + 2 m » — 4 71^) (5 m'* - 2 m n - 4 7i») . Exercise 53. 1. (a + ft) (a + c) ; (a c + rf) (a c — 2 ft). a. (a-.ft)(m-«); (a-ft)(4x-y); a(a + l)(a=' + 1). 3. (2x~y)(3a-ft); (;, + ?) (r - 3). 4. (x — y) (a — 2 ft — 4 c). 5. (a - ft) (5 a 4- 5 ft - 2) ; (2 X + y) (3 X - a). 6. (2x-l)(x«4-2); (ax - 1) (a»x»-ax -1); (a;-2y) {m — n) ; (a 4- x) (4 x — a). 7. (x + my) (x — 4y) ; (a — x) (4 a — 4 x + 5). 36 ANSWERS TO THE 8. (a — c) (S a — b) ; {a -\- b) (ax + b ^ -{- c). 9. (ox + Sy) (ax — bt/); (m ~ n) (n ~ p). 10. (m — n)(m + n -p); (2^/ — 3£c) (3?/ + x) (3 i/ - a:). 11. (c + 7) (3 a - 7 6 - 5) ; (ic - 2 2^) (:r - 3 2/ + 3) ; (a, _ 1) (;^2 _^ i^>^ Exercise 54. 1. (a-\-b + cy. 5. (a-b + c- df. 2. (a-5-c)2. 6. (3 0^2/ -4 ay. 3. («^ + 6 - c)^ (oj + 2)^; (^ a - 3 6 - t)^ 4. (a — 3 a:)^ 7. (m — ?i — ^ + a;)^. Exercise 55. 1. 10 ((K" + 1) (af* _ 4); (^2 _^ £c + 1) (a;' - a; + 1); 12(xy+l)(xy-^). 2. (a: - .5) (x - .06); (.^ + f) (« + ^); (3 - a:) (2 + x). 3. 3m27i(m + ^)(^ - m); 2 (2 a - 1) (4.0^ + 2a +1); (a^ + 9) (a + 3) (a - 3) ; 6a;«(£c + 6) (a: + 2). 4. (xy + t'^) (^2/ - t); C^^'^* + <^o) («'^ - f); \.(a + by^i]l(a^.by-^l 5. (a-x)(«'.-a:-4); 2(x'' + x^2){l-x)(2 + xy a(ha'+l). 6. (a^3+ f ) (a^«-^) ; (a-+o,) (^^-+1) ; (| a' --3 a«) (| a^ m^g a^). 7. (m-a)(w-7i); (a4.8)(a-l); (a-b)(4.a- U- 2)-, (la"" b^^ + a;^) (7 aH^"" + 3yi). 8. (17 + a) (12 - a) ; (x'- - if) (a;^" - i ). 9. (tw + w) (m — ?i— ^); «» (a; - 1) (a:^ + 1) ; a' (l + b)(l-b)(l-b + b^) (l + b + b^y 10. (20+a;) (19~a;) ; (a^-l) (8a^-l) (a^"'+ ^3'"+ ^2'"+ a-+l) ; l(x — yy^ + 5 ^2-] [(a; - 2/)^'' — 5 Z'^'"]. 11. (3a;-ll)(2a: + 7); 12(a:+7) (aj + 2); (x + y) (x + 7j-5); (f a: — J^??^7^) (fx- ^ ?/). FXEMENTS OP ALGEBRA. 37 12 (H-^3^ar)(l-rVa:); (2ar-2/)(ar + 3.v-2a). 13. (a" + b'x) (a^ + />";/); (a- - (i) (a; + 2« + i) ; (9x-^ + liajy - y-) {'^x' -2xy- y«). 14. b (a* + ^^) (</« -aH^ 4- 0') ; (a; + y)»; (x^" + aZ» + ac) (ar-^" — w ^ + ^ c) ; (ni -{- n) (m* — ?/i w + /i'- + 1). 15. (rt + ^— c— r/)(a-/> + c— </); (a— y + a; + 2) (a— a + y+2). 16. {x -^ 0) {x-\- a-i- b)] {x^" + 2 a) {x^" — a — b). 17. 2 (57/i + ii^ ± 1) (25r?iH 50//in + 25w^ if 5m qp 5w + l); (4w + 70(4w-+2m7i+77*2'), 3w(12w^+8mw + 37r). 18. (^c*af — a*) (/•'•(j^x" — 1); {7 p'' + 13;;^ + 11 q^) {Ip'- V6p n + 11 q^) ; 2 m {m'-\- 3 7^, 2 w (3 t/i'H n^). 19. (m— 7i) (2m— 271 + 1) (27?i— 271— 1); (m + 7i) (2??i + 7i), 71 (m -f- n). 20. (a- + a c + /> c) (« — a 6 — « c) ; (8 m* — 4 m ri + 9 n^) (8mH4m7i+97i2) ; (r)a;2+3xy+4y^) (5a:2-3a;y+4y2). 21. x(3a; + 2i/)(2a; + 3y); ^/'(Sar + 4y) (2a; - 3y). 22. (a:» — 3ic)(a:" — a^» — ac). 23. {m^^-\n^){m±2n)\ (4m + 37i - 3j9) (3w + 3/> — 2m). 24. (x»«" - c) (x"" + a + i); ' (X + y + 2)(a: + y - 3); (6ar» + 4 a; y 4- 4 y*) (5x^-4 a; y + 4^). 25. 3xV(3« + 2y) (a; — y); (m - 37?.)(m + 2 ti ± 4). 26. m\n\{ab—xy--^z)(,iv'b''-^abxy-\-?^xyz-\-?>z^-\-x'%f)\ (9 a- -f aJ" b^"^ — 11 i'") (9 a" — «»" ii*" — 11 A"*) ; (9 a*» + 3 a-i^™ — 5 ^»*'") (9 a«" — 3 a» 6^*" — 5 i*'"). 27. 2(3x-2y)(3x-2y±6); 2 (m + 37») (m-27t- 6a); a«(a''x^+4n^) («*x*-4a«7i2x-^+16a*7i^x*), a^(ax-}-2n) {ax—2n){a^x^—2anx-\-4?i^)(a^x^-\-2ftnx-\-4n''). 28- (x + 7y)(aj-4y + 4); 2(1 - 3 a - 2^) (// - x). 29. 7»77 (m + 7i) (m — ti)*; — a (a + m) (a* -\- 2am -\- 2m*). 38 ANSWERS TO THE 30. (3x — 4:y)(5x-\-4:y — 5a); (a — b) (a — c) {c ^ b). 31. (a -h c)(c — a)(cd — l)(c^d^ + cd -{- 1); (mn ± 8) (w^Ti^ qp 8mn + 64); 6 w^ (4m + 3 w) (w — 2 w); (a a; — 3 Z> y) (a — i/) . 32. (m - 2 7i) (m - 3 w + 16) j a;^ (3 a; -- 1) (a; - 1). 33. (m — n) {6 m^ + 5 mn + 7 n^). 34. 9?«.'(a2+ w^), 9m'(a + m)(w— a); (x+42/)(x — 42/ + 1), (x - 4 y) (a: + 4 2/ + 1 ) ; (a: - 2 a: 2/ + 2) (a: - 2 xy - 3) . Ten. 35. (36a:-132/)(18ar + 29i/); m(m+ 1) (m-1) (m^-w^-lO). 36. vi{m-\-n){m^ ■\-mn-\- n^){m^—mn-\-'nP)) {y-\-\){x—l) ix-y^-l); {x-\-2)Hx-2Y. 37. (?/i + 4)(m4-5)(m — l)(m — 2); (3 — w)(ww — w — 3). 38. (x» — ^ — a:-")2; {x-'^ -^ y-^) {x-^ — y-\). 39. 7o2a:(a;-2a)(2a:-a); (x-«+y-«) (a:-* + 2r*) (a^*-2r*). 40. a:2(12a:8-8a'2/H2l2/); a:i(4arJ±3) (16a:q:12a:i+ 9). 41. (a: - y) (a; — 2 ?/) {x + ari^/i + 2/)(a: — ariyi + y) ; (a;- + 1) (x"* + 2) (a:- -f 4) (oT + 5). 42. (2a + 3^') (2a-3/>')(ar-2a)(a:2 + 2ax-f 40^^); {m + 2n-^p)(m + 2n —p) {p + m — n) {p—Di + 2w) ; (a:- + i) (x- - i) (a:^- + ^V) {^"" + tV); («'" + *'") (a*" + 5») (a*" - b^) (a*"' - b*'' - 6 a^"* ^»2«), 43. (x^ + 2/') (a: -- 2 I/) (ar^ + 2a^y 4- 4 y^) {x^ - x^y''^y')\ (a;"* + 1) (ar^"* + 4) (a:^"' _ ar« + 1) (a:*"' — 4 ar^*" -f 16). 44. (^-\-n— p){m-\-p — n){m-\-n-\-p) (m — n — p). 45. 4 (a:"* + 2) (2/" + 4) (a:- - 2) (i/" - 4). 46. (a:"* + 1) (a:-" + 2) (x^"* + 4) {x^ - 2) (a;^"' _ a:*" + 1) ; (2 a:"* + 3) (4 x""^ + 9) (2 a:"* -3) (a;"»- 1) (a;2'" + a:"» + l). ELEMENTS OF ALGEBRA. 39 Exercise 56. 4. 6a«*x*; 2axy. 13. x {x - y). 21. 4rz-l. 5. Za'^^y^ !*• ^-^^y + y''' 22. x + 2. 6. 6x^2/-^z^ 15- 4(a-^). 23. x - 3. 7. 2a:iyJ. 16- 4;s(a:-y). 24. m - n. 8. 6(« + ^)- ^^- 2x-3. 25. or -2. 9. ^y^^. 18. x^'Sy. 26. x^^-e. 10. a:*(3x+2). 19. x - y. 27. a:" - 5. 11. 3a«x-12a». 20. x^ - x. 28. 2a:"-5. 12. x + l; a:" + 6; x + 3. Exercise 57. 1. ar« _ 3 X 4- 2. 9. ^^ (^ — 3). 2. a:^ — 2x + l. 10. 9m»(wi-l). 3. ar» + 2 X + t. 11. a: - y. 4. (x - 1) (x - 3). 12. mn{x^- 3). 5. x{x- a). 13. x^ - X — 1. 6. X — y. 14. 2 X — 5 y. 7. X* + 2 X + 3. 15. 2 n (m^ + 4 m y + 7 y'^). 8. 3 X + 2. 16. 2 m" x" (x« - 1). Exercise 58. 1. 2(x + y). 3. X4-2. 5. x" - 2. 7. x-2y 2. ar» -f X + 1. 4. 2 X + 3. 6. 2 x^ + 5. 40 ANSWERS TO THE Exercise 59. 1. x'-Vy. 17. x'' - 1. 2. x-y. 18. 71 (n + x) (w — x). 3. x-1. 19. x — 2m. 4. x-1. 20. xyi^-yY' 5. x-'d. 21. a-b. 6. '6x'- + 1. 22. a -{■ b + c. 7. x^ + ^y + /. 23. X" + 2, 8. x+l. 24. x-2. 9. x{x-^ b). 25. 2{x + y). 10. 2x'--^. 26. 7^2_^8x.+ l. 11. a" - h\ 27. n + 2. 12. a-h. 28. 3m(y'+4f-2y + S). 13. 3a:2« + 2m2. 29. x''-Sx-\- 1. 14. (m -n)(x- y). 30. 2(:r+l). 15. x^ + A. 31. x--2y\ 16. a" + b"^. Exercise 60. 1. S19axU/z^ 10. (n-xy(n* + a^x'' + x^). 2. lUm^n^x'z^ 11. x^-6x^ — 19x + S4:. 3. aea^^-^cl 12. 105xy^x^-y^). 4. 72 m^n^y^. 13. :r^ — 1. 5. 12aic3y4(a?2-3/2)2. 14. (3a:+ 2)(a^ + 2) (;r+ 3). 6. m'^n^ (x^ — y^). 15. {a ^ x){b -\- x) {c + x). 7. 12 a:iy2 (a,2 _ y2>) j^g^ 3mn {x - yf {m - n). 8. (x2-16)(:z;2_25)(^_6), 17. („4.^)2(^2_^2). 9. x(x + 2y{x+l){x-\-'S). 18. ^i^-l. 19. (a:^-/)(x2_^2). ELEMENTS OF ALGEBRA. 41 20. 3a«x(3a:-a) (2a: + 3a) (a: + 5a). 21. (x + 4) (X + 3) (X + 1) (x - 2). 22. (x-y)(3x-2y)(4x-6y). 23. a* - 1() 6*. 24. {a -I- ^) (w + w) (x + y). 25. (4 a; - 5y) (2 JB - 7 y) (x + y). 26. (x^ + y^) (x* - x^y*^ + /). 27. 20x2y(3x+l)(5x + l)(4x-l). 28. (a-f ^ + c + d)(a + ^' — c — c?)(a4-c — i— e/)(a + rf— Z» — c). 29. x* + xV + y*- 30. 6x2(x + 7)(3x-f 5)(3x-2). 31. 12x-(x" + 2)(2x« + l)(4x"-7). 32. abc(m — n). 33. (a — b){b — c). 34. ale {a — x) (^» — x) (c — x). Exercise 61. 1. 2x*4-x«-17x2-4x-f 6. 2 and 3. x* + 5 x* + 5 x'^ — 5 x — 6. 4. (x-2m)(x + m)(x* + m*)(3x2-mx + m«). 5. 2xy*(30x» + 95 x*y + 68x»y» + 32x«y« + 24xy* - 15y»). 6. X* - 14 x« + 71 x** - 154 X + 120. 7. (x2-3x + 2)(x* + 3x*-8x« + 40x-96). 8. ar* + 2x»-9x^-2x + 8. 9. 3(6x* + ar*~33x» + 43x«-29x+ 12). 10. 6 X (x - 1)* (.r + 1)«. 11. x*4-5x»4-5x* — 5x — 6. 12. 2x>-2x'-3x« + 3x*-2x«-3x'' + 2x + 3. 42 ANSWERS TO THE Exercise 62. 1. (a - ft) (a 4- by (a^ - 4:b^) (a^ - ab + b^). 2. x^" + 7 x^« — 10 x^'' — 70 a;2" + 9 a" + 63. 3. (a;" 4- 4 2/"') (x" — 2y^) (x^" — 2a;"2/;« + S?/^'"). 4. 2(x + 3) (2 X + 3) (x^ - 1) (x^ ^x^ + l)(x + 2). 5. xy{a^x)(b-y){2b-y)(2b-'-xi/). 6. a*"* — b^"". 7. x*" — 16 a**". 8. (ic" + c) (2 x" — 3 ^») (a;2» -j- a a;" — ^»2). 9. (ic + 2y (x' + 4) (X - 2) (x ~ 3) (x^ - 16). 10. (X2« ~ a^) (aj2« _ ^2) (^2« _ ^2^ (^6« _ ^6-^^ 11. Sx — y, (3x — y) {x + yf {x — yy. 12. 3^2 - 2 < 3 ic^ y (3a;2 - 2 a^) (a; - 9 a) (2 x + 5 y). 13. 2 a;" + 1, (2 x« + 1) (x^" - 1) (9 x""^ - 4). 14. The expressions are prime to each other, (a* -\- a^b^ -f- b*) (a + by (a - by. 15. a; - 5 ^>, 6 (a; - 5 ft) (x^ - 9 ««). 16. x^'* — 7 a;" + 12, (a;2« - 7 x" + 10) (a;^" — 7 a;" + 12). 17. c a; + *^ (c a; + ^^) (a^^ - c^)- 18. m^ + a; 1/, (w'^ -\- x y) (4: x^ — 9 y^). 19. a^m ^ ^2n^ (^2m _|. ^2 «^) (^2« _ 4 ^2m)^ 20. a;^ + X 1/ + y^, (x* -{- x^ y^ + y*) (x^ — 4 y^). 21. 5a;2 - 1, (5a:2 - l)2(4a;2 + 1) (5x' + x + 1), 22. a;" — y"*, (a;« — .?/"•) (a^^" + x^^i/^'" + y*-^), 23. a;'' - 8 a;* + 50 a;2 - a; - 42. 24. (x'' + 7x-^ 12) (a;2 + 0! + 3) (a; - 2)^. 25. (2 a;* + 5 ^2 ^ 3) (4 x* - 49). 26. x(6x^- 31 x* — 4x^ + Ux^ + 7x- 10). 27. a^s + 3 a;^ - 23 .t^^ - 27 x^ + 166 a; - 120. 28. 3 a;^ 4- 2 a;^ - 3 a! - 2. ELEMENTS OF ALGEBRA. 43 Exercise 63. 2 2x — 3y 2bxy' 37/i«aj«/' 2x ' « . n(x* — y^) 2 X + 1 m X — J ^ 3m + 2 .. ^ y"-» „ * + a? 3 + a ^- 3^;^^ ^(^-y>^ ^- °- r-f^' -2- 3m — 4 3(w + w) a + i + c a — ft — c + x 4m — 3' m — n ' a—b-\-c^ x — a-{-b — c 3 x — b m^ + n^ x"*"^ 4m*(l— x)' x+c" * m * b(a + b) 7n^n-\-x X — y x — 2 5x^4-1 X ' x + y ' a + 4' 9 x» — 4 X ' x—y—m c — d a — 2 b 2x^^ — 1 Exercise 64. ^- ^ + ^^ + nr3^' ^^^- 2x»-x-3 ' 3. x« + 3ax + 3a''+-^^; a; + 1 ^ "^ ^ X — 2 a' ' x*4-ar-12 4. (x-y)«; x + m + 14-^^— -^' X -|- n X — 10 5. 3x«-4x + 5 + ,f— i^; x'^ + y*-. 44 ANSWERS TO THE Exercise 65. 2m IIP' ^ 2(^H1). 2m« w^ ' ic + 1 ' m^ + ^^' 2^ — 2/ 2. . ; — ii— : : ;;• 5. — -\ 6. a + ic' m — n^ a- -d m« 2y« 7/1+^ = ' a:Ha;y + / a 2 mn^ m — 1 3.2. y2„ 7. ^i 7- • 8. (m + w)2' (m - /i)^ 2/' (3 a;^ - + //") ^,rn ^ ^,«y. ^ 2/'^» "• ic'" + 2/" Exercise 66. n — m n — m —(n—m) n — m h — a ' h — a a — h ^ — Q) — ay —(b — a)^ x — m—n b — a — (b — a) b — a , +-7 -x; etc. 2. m+ii—x^ —{x — m—nY —(x—m—n) m — a m -\- a — x n — a — b n — b^ m — b -\- y '' m — b -\- a m — x m — X (m — a)(m — b) n —y ' (jn — y) (n — z)"* (m — c)(n — x) (m — y) 2a: — y — 3 a — c + 3 4. ^ ■ 5. {a + b')(a — m) (b — 2 x)"* (a — c) (m — n) (x — y) ab(m — x) mnxy (a — b) (a — c) (b — c) —^y{^ — y) abc{a — b){a — c) (m — n) {x — y) {y — ;*;) ELEMENTS OF ALGEBRA. 45 Exercise 67. amny bm*y bmnx ab^y ^ c 2 b 5 a bmnybmny^bmny bmny abc^ abc* abc cm-\-cn am— an bn . Sa^bm 3a^m Sd'b — Sabn abc * abc ^ abc 3abm Sabm Sabm bmn Sabm 10 m n-\-20n^ 10 m^ — 15 mn 15 m — 3n 30 m 7* ' 30 mn ' 30 mn ' a ;* -4 (x-^l)(x^-^) (g - 1) (a*- 4) ♦• (x2-l)(x^-4)' (x^-l)(x^-^)' (a;«-l)(x^-4)' (m — 7iY (m + 2n) {m + n) m' (q -- 6) (a ^-f ^^ 2 (g + ^) (<^' + ^') 4 (g* - 6^ a* -6* * a* -6* * 120 m -f 30 10 n — 5 9m-6 *• 16 (w - 2) ' 15 (m - 2) ' 15 (m - 2) ' 7 2a«y m« - n« 2 a; (x — y) (wt + n) ' 2 a; (x — y) (m + ») * m (m' — m X -f x^) n a (m + a;) m« + x« ' m« + x« ' m» + x« * • a^ + xV + y*" a;* + x»y« + y*' a:* + x«y^ + y** 10 ^ (^ •- y)* 5(^' + y') 5y(x^- K.xy4-y') «' - y* • 5(x»-2^* 5(x»-y«)' 5(x»-y«) ' 5 (x» - y») ' X" -i- //" x^yCx'^-y*") {x^^ -t- y«")« 46 ANSWERS TO THE a{b — a + x) (x — a — b) (a -\- b + x) (a — b -\- x) (b — a -\- x) (x — a — b)' b (a + X — b) (x — a — b) (a + b + x)(a — b + x) (b — a -\- x) (x — a — b)^ x (a + X — b) (x — a — b) (a -\- b -\- x) (a — b -\- x) {b — a + x) {x — a — b)' — a —b c 13. (a -c)(b--c)' (a - c) (b-c)' (a- c) (b - c) x—1 6—2x 15. (a, _ 1) (a; _ 2) (a; - 3) ' (x - 1) (a; - 2) (a: - 3) ' 9-3a: 4 a; -12 (a._l)(a;_2)(a;-3)' (a; - 1) (a; - 2) (a; ~ 3) * mx — am x^ — nx {a — x){m — X) (n — x) (« — x) (m — x) (n — x) ax — am 17. (a — x) (m ~ x) (n — x) a;2-2a;-3 - (2 -f a;) (1-a;) (2-a;) (3-a;) (5-a:) ' (1-x) {2-x) (S-x) (5-x) -(a; + 3) (a; -3)3 (x' - 4) (a; - 3)^ ^^ (x^ _ 4) (x^ _ 9) (a; - 3) ' (a;2 - 4) {x^ - 9) (a; - 3) ' •- (■T^-16)(a; + 3) 2 (a; - 2 ) (x^ - 9) (a:2-4)(a;«-9>(a;-3)' (a;« - 4) >« - 9) (a; - 3) ' jB»"»4.3af" a;*"* — 1 a;^"* — 1 ^^' (a:*'«-l)(a;«'«+3)' (a;*'»-l)(a;^'" + 3) ' (x*'"-l)(a;'^'"+3) * ELEMENTS OF ALGEBRA. 47 Exercise 68. 6a»-16a-15 . IS b^ c +1S b c^-\- 9 a^c-\- 9 ac^-SaH+ Sab* ^' 36 a ' 72 abc 12 g» 4- 28 a;' -27 . x* + y* Aa + b ^' Sx* ' xV ' 3b ' cp-\-bm — an Sam ■\- 2han -\- Ibbn abc 12 an 9 a + a' + 12 . 6 (n — m) ^^ a — b 4. n > rrz ' 6. 7. 3an triTi n a«-3a6c + 6» + c* . a^' + a'c'-^*c« a 6 c a^h^ c^ 2a^ 2w^ 13. 8. 4 n — w — ^=' + «^ mnx 9. 31 47 19 16x ' 42m 30 y n a«6 + ft«c- • ac* 14. 11 ar* — y^ ' a;* — m*a5 m + 4 -2 m — 4 ' 4 ?«,* — m m4- w, 2ax abc '^"' m — n 8a;' — a* 1 . Sx m — 7 ' ar»- 9a; 4- 20' ^^^^' 4m(m«-3m + 2)* 4a — 66 5 "• 3(m«-n«)' (a;-2)«(aj + 3)* 16 0- ^^^ ' {m -{- n + x) {m -\- n -- x) (m — n — x)' 17 ^^y* . 1 -^ 2m^ •'^- a;*-/' x»-y*' ^'* m« + n»* g'+ 7 2 m* — 2ma;« **• ar« + 6x + 8* ^- {m^^x^Y ' 48 ANSWERS TO THE 21. a; 27. ^' + ^' ; 2y. l-a;« 22. 96 x^ 28. 1 (3 + 2 x) (3 - 2 x)^ X-1 23. 0. 3^2 _ ^2 _ ^2 _ ^2 1 29. (^ - a) (X - b) {X - 0) 24. 25. X + y -1. 30. c — a — b (a-c)(b-c)' 26. 1 31. 2x» . Sa + x 2x + l x^ — 4 ' a + X 32. 1 0; 2x2- 64 9a; + 44 33^ ^^ (rn^x){x^2)' + x« Exercise 69. 1. J3 ^ 3.m + « 8. m {m — n). ' 10' 2/'" + "' 2. 8 4cie* 9. 3. 1 . 3a^ -1 -2 10. x^ a^ y"- b"" a^"^ x" V 2/'' 2 - X - ic^ ' X - 4. aj+l . ^ 11. (a _ c)2 - 6^ x + 5' rr^ — mn 4-71^ a6c 5. 12. 1 6. m 13. a;«-l m — n a:«+l 7. m^ — n^ . X + 14. ^2n _ y2m 2 (a;2» + 2/''-) ELEMENTS OF ALGEBRA. 49 Exercise 70. acmx 3 4 xy . xy 3(a-^b)\ a*-^2x^y-\- 2 xy' + y' b{a-{-b)' x*^3z»y-^4x^y^-Sxy^-{-^' x + y . 1 ^ b — X m*— 2m4-4 sb*— y" * a — x 2 a; — 1 g + 6^ — c **2x — 3* ' a-\- c — b' "• »* - n*x^'+x* ' a^ + a-' + l 12. 2a;V-4a;V + 2xV. Exercise 71. a^-b^ 10. 1. 13. 1. a — 12 >! «rfi+i (J a: - y)« ^ , a "* « 3. i ^lJ- . 4. X + 6. x-y 1 1 5 (a + ft) (g - 2 y) X x* • (a -I- 1) (X + y) ' „ J 2a6 b a 7. 1. 8. 1. 9. 1. 1^- ^'-2+ -,. ^ m5 77tt-r,r-^; 8^^-«- 17 ^^ . 18. lOa. h^ J8 6-y" ■*-^- aP^c* 50 ANSWERS TO THE Exercise 72. 03 + 6, a ■}- ni ^ an -{- cnx ^ mnot^y — 3ic^ ic — 6 ' h — m^ cm + cnx^ m^ n^ •\- 2 in n^ x h ^ m -\- n ^ 7ri^ — h n ^ 2 m a ' m — n a 7i ■\- b in m — n x^ + 1 x'^-U 1 3. —^ ; -t; T^ . 5. 2x ' x^-lU - 2j(rn-n-p)' 2x^-1 .. . r (a+x)(a '-x') x'+l . -, x'-x+l 1 + a;2 a;2 _ 3 ^ ^ 1 4 7. 8. 14-a: ' ic^_4£c + 1' 3(1 4- a;) * + 2/ . am-{- adn _ £C — 2/ y ' bni + cn-\-bdn x + y m -\- n wn 9. a + a;; 1. 10. -2- 11. • {m — n)^ m -\- n Exercise 73. ^' ~8F' 16x''y^'' x'2/ ' 256x12* 160^"^^. (a; + yY . rn'ix-ijY ^' Slm^nf^' {x-yY' n'ix-^-yy' {x^ - i/y . (a - bY . a''(a''-15a^+75a-125) *• (m + nY ' (2c^ + 3 6)2' ic^^ («_5)2 8 a;'^ - 36 a;^ y" + 54 g;B y^» - 275/»" . /^y'+" 5. . . ,..; Tin » I ; _ r ai m^ x^ 1 ' ELEMENTS OF ALGEBRA. 61 Exercise 74. x^ Sm*n* 6ac» «* m 1. ^^*' X ' 7x» ' 36*-* 2. a»» ' b^ ' 4m?i^ 5aH* 3. L*a*a''- a~* . aixi 5. x + 1-^; a« -hi 3i/-z' V^' 4. aj + y . / .^7 * 6. a ft. 3 -r*'- 2V(a;-y)-' ^ Exercise 75. 1. 1 . x-S 1 + x^' x + S' 5. 46; g. 2. Ty^ — Axz 6. ]§. 8. 0. 10. -H 6««-7xy* *; 0. 9. 8. a Sx*y-2» 7. 11. -f 3. 9 x« - y«« * 15. (a - cy - b\ 17. 1. 4. aj + 5 16. a:2 4- 3 x + 3 - 12. ":+"%.»; ^:-i + K 13. m« + — 4- ^ + i; :l _ ::^ + !^^Jf _ '^, n^ n"^ v}' y' ny"^ n'y n» 52 ANSWERS TO THE 18. 19. \n inl \n inf \ a/ \ a^J \ic** y^l (a^ ax x'^\ l?~b^'^ yV '■ 5^(y'" + l)(y" + l)(2/" + l)(y"-l),orx*»(2/^"+^) 20. ic. ^ "~ y ''®- ^- 2^ 28 (a: + 4) ^^' ^2/ * 3^ ^ 36. ^-±^ . 41. 3. ^c + ac -*- a6' 42. 1. 2b (by — ax) 22. 23. («,_,)(5_c) 24.-^. 25. 1. 28. 0. 30. 1. 26. 2. 29. a;. 33. 3. 27. 0; (a«-68)2. 34 2. a:(a^ + l) . 1. 1 45. '^ ^^' x' + ^x + r ' («^ - ^0 (^ - ^) 32. '^; ««^'^c^ 46. 4- o#. ax (S ax — 5 by) 40. 16 aH^ 43. 2{a + b + c)^ + a'' + b'' + c'', 44. 1 c (a — c) (b — c) ^1; 2 a ELEMKNTS OF ALGEBRA. 58 Exercise 76. 1. 10; f. 4. 6; -|. 6. 2. 8. 4. 2. 20; 5. 5. i. 7. 2; 6. 9. -2; 14. 3. 2. Exercise 77. 1. 1; 6. 5. 8; 0. a 4. 11. 3. 2. 1.3; -2. 6. -4. 9. 4. 12. -.04; i. 3. 0; f 7. 1^. 10. %. 13. li; 3. 4. 2; 0. Exercise 78. 1. n — ;— ; wv(l-3a). 5. a-b; — — r 2a 2ah ^ n * ^* 36 ' a + ft <? 1 ft I ^ 3. 4m + 8n; -T- 8. 0, -; ft-1. 'aft c a • w» 4. j^. 9. i^ + 27^- Exercise 79. ^ c , ,, 13. H« + ft + 3). 3ft 2a» 2. — • 3. 4. ^^ 28«+5ft«* 12. 5. 5 ^(a-6+c). *• ^* 16. —mn + mp-{'np.l^. 0. 6. 0,-; 17a. 7. 3(n-l). 11. 2. a a* 2 oh o b — a* a + ft 6. - "^ . 3 rf» _ 2 c« 17. 36. 21 19. — To— TJ— • 12 c a 7. 0. • 18- ^• 1 n (n^ 4- m') 54 ANSWERS TO THE Exercise 80. 2. 31. 3. 84. 4. 36. 6. 1^^ days. abc ab -{- be + ac days. 8. 10 days. 2abc . 2abc . ^ 2abc 9- —r-, 7-T-' -A., —^ 7 days; B, , , , ab-\- ac + bc ac + bc — ab "^ ab + bc — ac 2abc days; C, -^j—, r- davs. -^ ' ^ ab -\-ac — bc 10. 48 minutes. 12. 16 miles. 13. ^ miles. b-{-G 15. 150 miles. 14. 742^ miles. 16. 56 hours; 84 and 70 miles, respectively. acn . . acm .. _, abn 17. - — ; hours: A, - — ; miles; B, miles. bn + cm bn-{-cm bn-\-cm 18. First kind, — ^^ ; second, — ^^ • m — n m — n 19. ^^. 21. 69. 24. 160 miles. 2 m n {2 m -\- n) 4: m^ ■\- 4: m n — n^ 20. -, — 2—-^, =^-^ days. 22. $1200 in 5 per cents; |2000 in 6 per cents ; sum, $3200. $100 ^•m . ^100 b(n-m) . ^ 23. ^ in a%; — ^ ^ in c%; am -\- en — cm am -{■ en — cm $100 b n sum, am, •\- en — cm. ab -1 , a(3w2_^3w + l) 25. miles an hour. 26. — ^^ ^!^ — — - . b — ac (ti + 1) ELEMENTS OF ALGEBRA. 55 Exercise 81. (a: = 24, I « = H i3, = 12. "• 13,= 7. 3. }^ = 2J a. J^ = 24. 12. j^ = l^ Exercise 82. *• ■jy = 3. "• (y = 12. " U = 3. _ (« = 3, (a; = 38J, ja; = l, '■ ly = 2. 1y = -21J. • (y = 4. 6? SI 1. '•iy = 60. *• iy = -6. "•iy = S ty = 12. '• |y = 7. ly = - ,x = 19, r X = i», 1y = -i- 1. Exercise 83. x = -t, . (1 = 21, „ f« = 13. )y = -i. 1y = i5. ty = ll a. J'' = J' c-r^fi 10. 1'' = ^' (y = 2. (y = 6. )y = 5. fx = -3, <x = 10, 11 j* = 4' '•iy=12. '1^ = 11. "•1.'/ = 6. 4 (* = *• 8 i*= ^' 12 i' = 216, '••ly = -2. "•ly = 12. ''•ty = 144. 56 ANSWERS TO THE Exercise 84. ^•iy = .02. "• 1^ = 12. '^- 1^ = 12. 3. .P = -2' 15. r = 12, 27. j^ = 8, b = -3. |y= 6. (^ = 7. fa,= 10, ra; = 7, p = 4^, ly= 8. U = 5. ^"- t2/ = 5i. 5. 5=^ = 25' 17. 1^ = 10' 29. r = f «-L=4: ^°-{,=i5: ='='-i,=io. ^•{, = 12. ^^•{, = 1. ^^■], = .08. f* = ^' 22 jx = 20, (a; = i, iy = 2. ""• 1^ = 21. ^*-iy = 2. jx = 60, 23. r = 2' 35. r^^ 10 11 12. 1^=^' 24. r=^' 1 y = 4. * y = 7. ELEMENTS OF ALGEBRA. 67 X = 2. r (y (y X y X y X y X \y = 6. 7. (y = rx = \y = X = y = i §. -1, -2. -44, 6, 8. 12, 6. am—bn Exercise 85. 9. r = ^' 15. x = h 11 12 13 <x = . \y = l {^ = h ' \y = l (x = -l iy = h (x = ie, \y= 7. <^ = h \y = h Exercise 86. 16. i^ = J' (x = i, 17 18 19. 20. 21. x=l, 1. (y = « = iWr, J" I- x = y = ar = y = an — bm a^-b^ ' nq — mr I a* + aft + &• Iq — mp ' 6. - ^- a + 6 Ir ^ np Iq — mp' aft l^= a + ft- be 7. - \x=''^p:. qr-p^ ac ., -P9'-*** a«4-ft»' l^-,v-y « + fli ax^ — a OiA + afti ' 8. i X = Sf Oi — a* *,-6 1 — a Ox 1/ — *. axb •\- abi Ox — a* 58 ANSWERS TO THE 9. 10. 11. 12. 13. < 14. < am^n h + b a a — b ,2 a' {a — c) y = - a — c nr X — — , c X = c{a + b) 2^ c (a — b) 2a 2n m -\- n 2m m ■\- n 15. f m — n X = m-\- n aai (a — ai) 16. ^ " -r «i I _ aai(a + ai) 17. 18. 19. 20. 21. 22. 23. r \y X = m -{- Uf m + n. a be I- a + 2b x = ~ } m n 1 — > n L^ m r a; = a -|- 5, \y = a~b. (x = m + nf \y = m ~- n. a'^jb- a) ^- a« + 6» ' y = a^ + 2b^-ab b — a 24. { X = zfy mn — 1 y = 4-1 mn 25. ( X := m — Jlf \y = n — m. ELEMENTS OF ALGEBRA. 59 Exercise 87. ( x = 3, x = l, 1 x = -, a "1 ( y=2, .z=b, x = 10, 4. ^ f y = 2, z = 3. x = -5, 7. < 1 1 "• y= 2, ^• y = 5, 'x = 20, 1 .2=3. I « =5. 8. < . « = 30. ar = 2, ( -:r= 7, rx = -12, '■{ y = 3, .« = 4. M .;s= 9. 9. ^ y= 6, U = 18. a^i 5 > 2 ^-a6 + *c 4-ac ~ ?/l + 71 10. < abi ? 9 12. < 2 ^~a6 + 6c 4- ac ^ m-j- p abt 5 • 2 [ a4 + ic + ac . ~ w +/>" par = 4, rx = -li, r' = i, 14. . y = 5, 17. ^ ;y = 2i, 11. . [v = 3. \x = a, U = 6^. a 15.. y = b, 18. . y = -3i, » = 2' z = c. U = 2^. 13. V /=2- 16. . rx = l, 19. i fx = 7, y=9, U = 3. 60 ANSWERS TO THE 2. 3. iy = 3, 2. 13, 5. y = a + b 0. 4. r z = a, b. 6. Exercise 88. (x = 2, 13/ = 3. ^ = ^^^475-2' a^bc \ X = 5^7^, ^ ~ abi + aib 2 a aib ^ ~~ abi + aib 11. 10. mn 12. 13. 14. ^ — ^^ 2/ U — 3 5- y = m X = 15. ^ 2/ = z = f._ m + n- Sp w a — n n b + 3p- — m m c 20. f _ (a + ^) w + _ (a + ^>) ^ + 2, 3, 4, 5. ^m "^X" 16. VJ.\y = 2 m ' mn -\- mp -{■ n p — n^ — 2 p"^ z = 2(n' 3np — mp - p') m n 2 (n^ - P^) 19. y = z = 18. f = ELEMENTS OF ALGEBRA. 61 X = 21. ^ V = mnp mn + np — mp 2mnp z = nip + np — mn 2mnp VI n -f itip — np ar = ^a + * + c) — a, 32. X = 2 mnp mn -f np- mp y — 2 mnp mp + np- mn 2 mnp mn -\- mp — np 1 x = - t a 23 rar=|f (a + ft + c)-a, i I z = § (a 4- * + c) - c. 1 27. f^ = ^^* + W* -I- 7i*, >8 30. 31. 34. X = 105, y = 210, z =420. a; = l, y = 2, « = 3. x= 7, y = 10, 2= 3- = 7W,* 35 (x = m''-n", 2. J. 3- h h 4. ^. Exercise 89. 5- A- 6. a-\-bm b + an mn— l' mn— 1 10, -1. 7. 16i, 15. cm — an 10. Corn, i; oats, f . ,, /t bn — dm 11. Corn, ; oats, . oc — ad 'be — ad 12. 180 lit 2 for 3 cents, 300 at 5 for 8 cents. 62 ANSWERS TO THE 100 a (b c d - 12 en - 12 np) . 777 r at a eggs for m cents, 13. ^ d(b m — a n) °° ^ 100b(127nc + 12pm — acd) TT, — c at b eggs for n cents. a {b m — an) °° 15. 245. ^l^L±_?!l!!) ^^* " 2 a - 81 ■ 20. 853. 16. 891. , ,, , 19 ^^^ + M11-^) , 21. 315. 18. 39. 11 — a — c 23. A, 120 ; B, 80 ; C, 40 ; altogether, 21^9^. . ac(n — m) ^. ac(n — m) 24. A, ^^ ; B, ^ . nc — a mc — a 25. Arithmetics, 54 ; algebras, 36. 26. Crowns, 21 ; guineas, 63. ,., nia^c — a Ci) . n(aic — a Ci) 27. Crowns, — ^ : guineas, — ^^-^ ^ . a^n — am cm — c^n 28. A, 50; B, 21f. 30. Stream, 2; A, 10. . mn ^ mn 29. A, ; B, 31. Stream, a a -\- n — m mhi — rrixh . mhi -\- m^ h 2hlH '^hK 32. 39 miles, 8 miles an hour. 33. 5. 34. Going, 4^; coming, 1\\ stream, 1\. hh . ah ^ m (a" — b"-) 35. Gomg, ^-p-^; coming, ^-^; stream, \^^^ ^ m{a-\- by 2abh ' 36. Sugar, 5; tea, 60. r^ 100 nOO (71 - m) -hm'] 3,J^-^-' TW^) ^ 100 [am -100 (yi-m)] ^ea, ^^ (a - 6) ELEMENTS OF ALGEBRA. 63 38. Sum, $1000; rate per cent, 6. 39. Sum, dollars; rate, -— r— • 40. First kind, 14; second, 15; third, 25. 41. Fore-wheel, 4; hind- wheel, 5. 42. Fore-wheel, ; (rnr-'sn) ^ ^^^^ hind-wheel, ' (m + 7i) (cs + cr -{• ar) b(mr -8n) ^^^^ \$ + r) {em + en -\- an) „. , . . . bnti — bim aim — ami 43. First kind, — { \— ; second, — r j- . aib — abi Oib — abi 44. First, 3} ; second, 3. ^. ^ a(m-\-l) . a (n -\- 1) 45. First, — ^^ T^; second, — ^ :r * mn — 1 mn — 1 46. 3^8, 80% ; 4's, 125%. bd -\- np bd — mp 4a A, 55; B, 105. A ^{n — b) (m -\- n — a) c {a — n) (m -\- n — b) *^' ' m~(^r:^) ' ^» 7^7^^^6) 50. Sum, 500; rate, .04. * ., o bm — an ^ n^ m 51. Sum, —7 ; rate, b — a bill — an 52. Pounds, 60; cost, 28. 53. Larger, 5.678; smaller, 1.234. 54. Smaller, :j -; larger, - 1 — no 1 — a 64 ANSWERS TO THE 1. n^^ i. 216 a« 1 3. ^ a.^ + 1 4 ™n + I . 1 . . Exercise 90. a' 4- /^«'-*'Y "•''*' u + w ^- ^ 2" ' a' (a'" - b*") 11. ^"'f'.^ + lZ;^'''-"*-!; 1 1 16 16" a"""* a;""' y^ ' • m m > 1 27 • Exercise 91. 13. aVa;^ 6^2 6^c2«'. 14. 2". 15. 22'". 16. (a;2-y2^-3«j 2«. 1. ^ ; 1 ; ^a;^ 1. 8. V-{a-2 x) 9 ^\17(l + a) a»' _i_. L_. 1 9. a;2«; 1. 1 - „- 1 10- (^ + 2/r; ^"-^'^ -6 3- ^r-;;r; va»; Vet'"; - «- 11. _1_ /^\ {m + n)^ 4. — 4 a/2 ; ^a\ xy. 6. 3"; (a -5)"-; g- 13. ^a ; 2«'. 7 ^\'/"^. ^' 14 a- a;« + ^+-^"- ELEMENTS OF ALGEBRA. 66 Exercise 92. 1. 9; X; 4m-*; -^; ^. 2. IG; o^^i'^; 516a»''6»»-^c»''-«. 3. 343 aV"*"*; (27)'»6»'-« Va*"'; a:*- 5. |,; y»^-^-; 25. ^- S ' 1 5 V^' 7. _^L^j (25)'"'»a'''r***'"c'^'"^"S J^4. 8. a:"'"; a^'^"- Exercise 93. 1 ,1 ,1 (b-\ ^- (^-Y <""">' (^»' (''^^'' (;^' i^'i Va-'ft-'y-*'' _1_ « m 4. -fo^; ^iO^^ yaHW', yJ a^^ b" ; \ ab^-, ^ x^ i/\ 6. Sa^bc^', 5a-^b hr^; mlnij ab^; x-^t/-^; ^"*y^; «'**• 2 • 1 1 1 5 _ 1 ^- 5a-«ca:^ a-J^!^ -^^^ xjri^*^ Sa^J^V^ ^""^' rt* 1 8. «*; jgj «*"•; a»; -; al. 66 ANSWERS TO THE m 10. -; 7^TT-^', 1- 20. -t; 1; 1. 12. |;;16.^m;^; 22. 2«%^ M , ^^ 23. I; 18X10^^' \/nr J/7. ^"^ 3' 2 »i m t t 1 6/- a; " Veto * "? ^ ± 1 27. a'* — a2»^»4« ^ 523 a^ Z.2 ^ 29. £c2n-2_ 2m- 17. ao''; fw7^ic. "^ t m t 11 . . 32. a** — 1 + a~'* ; n^ y^ — n^m^'xiy^ + mTn*xiy^ — m^n^x^y + m^n^x^y^ — m^nrxy^ + ?7i7a3s. (as" + &-'*)(ai'* — b-''). b^ Ab^x 6a;2 Ax x^ , ^ 12 8 34. ^ + — T 72 + 745 x^ — 6x + — + -^', a* a^ a^ ab^ b^ x x^ 5n . 5n a 2 _ 5 ^-^n _^ ^Q ^-^n _ ^Q ^^n _,_ 5 ^f , a 35. 2! X 3^, 2i X 3^, 21 X 3^, 21; 24. 36. 2§ X 3^, 2i X 3^, 2i X 3^, 2i, 2i X 3^, 2§ x 3*; 576. ELEMENTS OF ALGEBRA. 67 37. ai + i; 26i- a-H\. 43. a'-'-^+a; 2^^' 38. 2a-\-a-\. 44. 77» ^ 39. x-^x-\ y' 1 45. xi;2. 40. X ,,-^ ,; 2»«% 2. X 46. ♦1. 3-2a^-. i^m" 42. x-^ -. 1. 47. (5a:^-3yY; m« - -w«. Exercise 94. 1. V^i; \^V"; ^64; ^i*""; ^8"^; ^27 aH*c^; '^/W^', ^; ^^; ^i^^. y/i; y/J; ^/2^; ^F; a. Vi; ^i; V800; VH;_v/^¥^; v^^^. 5. VW^^l^^, v^ii^^l; V/^- 8. 12 V2; 15\/6; -9v^; | \/58; \a</Va. 9. |V2; f\^4; ^-^^20; 7^3; iV5;iVl5; i ^32. 10. i'^'24; VV6; -3a6^4^; 2aJ6«'^; ;r^,\/3a*na;. 11. a*i-V^c; -V^; — V^^aJ. 68 ANSWERS TO THE 12. ^ ^6 ; 9 a" b^"* \/Wl? ; - ^2ax'y'; x y"" ^yx\ 13. (x + y)Vx — i/; (x — 4:)^a. 14. t>(^ + y)'''' 7 ^3n^2,n^5^ — m 15. 72; 3; 2. 20. fa^c; a^oTx. 16. 17. 21. 22. 1. 3 2 2' «• 18. 10 a m X. Bab. 23. 24. 2 a. 19. a</4; 2a«Z»'^8a2i» Exercise 95. 1. v^8; ^9; §^1296; 3^64; ^^»; ^8; J ^16. 2. V3; tv^64; {^«^'; •'^?. 3. -v/a^?"; y/o^^; y/a^ SJx'yl', ;/a-; y/-^; :.'Y«'- 4. v^l25, V^m, V^l3; v^i024, v'625, y'MS; -^^Gi, ^81, ^6. 5. ^2, ^2, ^^2; ^49, ^625, ^2l6; ^a«, 3^^°. 6. ^^, ^^; ^a^, ^^^ ^^; ^^, ^^^ 7. V< ^625^, ^27^; VW^', V27 b^ V^^x^ 8. a:^< ^^«"^; ^< ^< ^^, ^^^^^^^. 9. '^a', W, ^Z^; 4^625^«, 2 v'8T^^ 10 a ^729^ ELEMENTS OF ALGEBRA. 69 Exercise 96. 1. 2Vi4; eVH; 6^4. 3. V^l v^4; v/f. 2. 10 VS; ^ViO; ^2. 4. ^Ti; 1.6; V^. 5. V3, -^7, ^4; 8 a/2, 5 a/5, 4V7|. 6. 4^/7, S'C^iO, 2V^21; 4v'ii, 2^l3i, 3^. 7. 3V2, 3^5^, t>^; 3\/8, 2^/8, 2^21. Exercise 97. 1. 8\/5; tVlO. 7. c (6 c — a) \/2 a * c. 2. 2V3; 6^6; | v^2. 8. mnx\/mn^. 3. -a/3; 17^. 9. 2bVS~a. 4. 20 J a/3; fA/15. 10. - ^'^^~J\ 5. ^a/IO; 0. 11. f|A/6-^14. 6. — Vva*; 2v5. 12. — ^^ s ^a/wx. Exercise 98. 1. 18; 18 a/7; 25 a/5. 6. ^^^486; v^ll8098. 2. 2; 14 a/G; i >^12. 7. § >^ ; J a!^ ; 14 >J^. 3. 9v^288; 10 (3- a/3); S. 8. -729; ,— 6(l-3A/3 + 3ViO). 4. 2+ A/6 + 2VIO; 120. ^ 5. vTO; T»B ^2592; ^500. 9. A/ lOO; n-V^; 12v^2. 70 ANSWERS TO THE 10. in n ^72li^x\ 19. 5 ; 4 \/32. 11. m^m^7i^x^^', 6\/^iH\ 20. 4; '^3. 12. v'72^; 2«m"a;". 21. \/^- Vn-, yVv^lSOOO. 13. ^V2; -12-1 V^. 22. 2; -v^9^^ tt X 14. 9ViO-12 ^32 + 6^^3125-8^/10; ^^4 + 2 -^6 + v^9. 15. 4(6VlO-l). 16. ^^128 - ^^2187 - '^972 + -v^O + v'i - V6 ; 12 ^^27. 17. - 1 ; 'v/y - 2 ^^2. 23. 49 a; - 9 a. 18. 2 ; - 7. 24. 1 v^288 ; | ^^80 ; 20 \/b. Exercise 99. 1. 2(a/3-V2); ^5(2^/5+^6); 3^/2-4; 3(VT5-VtO). 2. 4+V2; -l^yV^ ; _^(3 + 2V6); t^re. x—2'^xy-\-y '\/xy{?»x-\-^\/xy — 3y)^ a-\-^d^—x^ x — y ' y{x — ^y) ' a; 4. 2 i«^ - 2 X V^^^^l - 1 ; ^V^(^^- h) { a - Vb-) a^-b 5V5 - VTO + 5^2 + v/8 2 a \/a:'- '"«'-« 23 ' 3 a; s 5. 14.14; 7.07; 141.4; 11.18; .1732; .2236. 6. .707; -.236; -1.266||; 1.216§; .268. ELEMENTS OF ALGEBRA. 71 Exercise 100. 1. 3V3; |V2; '^3. ^^ axV^-^Vy; — ^^^^^. 3. 3.3; ^VO; i ^486. "^ "" 4. 2^3; ^54; ^18. ''• V ^^-^^ ^ ; - ^1944 aV^ 5. 6^5; i^96; 2^/2. 13- ^^'^^^^5 1 4- V^. 7. 5 V7 - 4 VC + 2 V5. ^3 5 - V6; ^ '^a^^^ .- le/-^ 3aa; 8. .2; a^6c; --^aft*. ^ .-^^ ^ , /^ * 16. 1 (4+ V15); r)+ V<. 17. ^x + V^^ + Vy. 9. !!Lzl!^; 1^96 Exercise 101. 1. ^^; V3; 2V3; V^8; 2 A^^i 2. i>^; i^J^; V«-^; 2. 3. 4^; ^n»; 2^'/2; V2; 2 15^2. 4. V^; 3^; H; 4«^3; ^ ^a.' 5. 8aVft; ^V^i^^^^; v'-^; 32a'bc\/2abc, 6. 3a; 648^/^a:^ 2a«; ^^ 7. r,; ^(«-rr; 1; ^^^^ 4 aC 72 ANSWERS TO THE 8. JV3^; a'"-V«; va''; -4- 1 X, 9. -\^n^—'-'; "''^(aH'^)"^^'-'^^^; m'n' ^/n\ n 1 8/-r- 0^^ iC*' V^^^ ^^ 10. -,; Va<o; — ^g 11. ^.; ^2/(^ + 2/)^ 2a^^3. 27 12. x^ + 2;r27/2 + 2/'; t + 4V6; 11 V2 + 9 a/3. 13. 2 - 3 ^y + 3 v^3 ; 4 + 2 V2 + 6 -^4 + 6 '>^32 ; m' '\/') m^ 3/— , O 6/- iiin 14. _^ ^6 — 4a" + a^"; v<*^ — — ^58m- 16 15. 5 10 10 5 ^12 + a« + a^ ^^ + ^e + ;;r2+:;F+-4 + l; «*- + 4 a«- 5 + 6 a^- 6^ 16. ^V2m7i + |--2^m7i2 + ^^8m7iH404- — ^Sm^Ti , 32m 3,—, ^ , ... --- , Q>A:m^x^''. Tn^w w 17. ^ + 1-3^2/. 19. ^ + «^^2/'- 2 2/^' Y2 18. 1 — 2" ; 3" — 2\ 1 6"/;:;57 20. ^^a^--2V;i^- + V^ ELEMENTS OF ALGEBRA. 73 Exercise 102. 1. ^9; 1.2; 9; 1; i; |. 2. a»v^2; mv^G; i^768; 4. 3. ^^5^; ^{/243; 6^3', 4 Vm (m + n). 4. -V^^?^^; i^xTy; e^^^^v^a. 5. ^64, ^?^, ^l25; ^6561, '^EV^, i?^15625; 15^2?; 9^4. 6. 6^*64, 6^16, 12 ^8, 10-^4; ^a\ ^6", ^^\ 7. Va, 2V3; 3v^2, 3^2; 2v^9, 3^9. 8. iV5, a/5; V^5,3a!^5; Vtr VlO, t^^VIO. 9. §a^9y^2ftv^96■^^^^'9P; ^ ^50, ^ ^50. 10. l</% \^% ? ^9, ^; 2 V5, ^ V5, 20^/5. 11. V2, V2; 2a;V2, %bW% « V2. 12. i ^, 2^6. i ^6, i ^6, i ^6; 2 v^, ^2. 13. 4^^, 3 \^3, 3^/275-, 3V19, 5^8, 3^9. 14. 2 ^8, 5 ^2, § V3, i V5; ^H, Vf "• ^^^VS^» -^^5^(2i^^,aV(8^^. 16. 16 V3; 7^7. 20. 8^4; v^3. 17. 4>^l6l25; 3 15^4; ^624288. 18. 3^/2; - ^3. 21. H\/15; T^y. 19. 24v^4; 1. 22. i; 6. 74 ANSWERS TO THE 23. |172; 0. 34. 1; i; ^^ ^^. 24. -/s'^^; ly^. 35. -^^V^; 0. 25. 156-24V4. ^^ cm _ 36. 11; — -. 26. f(7-2VlO);iVl5. ^"^ Vw/^ + «^ ^^ — <^ ^ 1 27. ^l:::_lj^l^! — ^-. -V^«z/--- _j V2+iV3+^V3o. 77t y 28. ^ v^lOSOOOO ; 204.8; ^a'h'', 29. 47; 27. V^. icB - 2 30. a^-i + ljar^y-t^n+D.j/^ ^ ' xl + 2 31. a^^. 39. |V2. 32. cVc; ^a. «,y^2o X 315 ,.y/^ 33. c ^c ; h '^Wb, V 2^^ 38. 1 ; Va (a V^ + 3 a + 3 \/a + 1). Exercise 103. 1. 3V^^; 2\/^; 2V3^xV^; 2aV^; iV^. 2. 7a"^»V=^; 3'^=!; a^^/=l; 2'^-^. 3. -1; -V=^; -a/=3; +1. 4. +V^; +1; -V=^; -V^=^. 5. _4Vin[. 8. 2&(2a2j-l) V-^. 6. 4 V^ - i V^3 - 3 V^^ ; 0. 7. a2(8a + a2_^y'ZrX. 9. -3V2; -36^/3. ELEMENTS OP ALGEBRA. 76 10. -48; 2,^/^-i, 16. 6»-a» 11. ~GV<i — ^9. 17. V3j AV3. 12. 39-2V^; 21. 18. i; 2V2-\/3. 13. 2; — 4 — 5\/^^- 19. 4; - Va — V^. 14. 30 + 12 V2. 20. 1 - V^; 1 - V^. 15. (a - h) V^^; a + «. 21. i A^2625; V3. 22. 1 + V^^; \ V^^ - i a/3. 6 + 2V^ + 3\/^~a/B 24 + 10V3-15V6-8V2 ^- 3^6 ^ 43 5 lV-1. ^ / — T- a* — 2 a V— « — a; 2 \/a^ — a — x 24. 1 — V — 1 1 7. : } • Exercise 104. 1. V5-A/2; V3 + 'v/2; 4\/2-3; iVB + 1. 2. VI0-2V2; V6 + V5; V7 - \/6. 3. V14-1; 2Vll-V'3; V^-l; V6 + 2. 4. 2-iV3; V^2 [a/3-1]; a/5[V2 + 1]. 6. m Vw — V^ w; a/'J' -- a/2. 7. m — n — 2Vmn; /v/^('y/2 — 1). 8. 2-V3; i(V5+l); a/5 + 1. 9. a/2 + 1; V5-V3; V^(V5+V^). 76 ANSWERS TO THE Exercise 105. 1. 11; 10; -f. ^^ (x = 7^-^VTS, 2. 4; 15. * (2/ = |A/i5. c — 1 9n^ 12. —; 34 m ^- ^^' "^'^^ f 2V«(V«+ v^) iC = a — b ^^ I "^ I) — a ; 54. 16. ^ I ^ ^ 2 (Va6-l) 5. 2; -,V '^ ^-^ 6. 192; 3^. ^^ (^ = -li, 7. 4. 8. 1. f _ Q^ — ^ 18. ^^-l-n' ^^' 4m '^' ^i- r m + y.-^ - } 13. 15; 6. ^^ I Vm 14 ^(^ + ^) . h Vn Exercise 106. 1. 0.7781; 1.8060; 1.1461; 0.9030; 1.0791; 1.1761; 1.9242, 2. 2.5353; 1.2040; 2.3343; 1.4313; 1.6532; 1.5562. 3. 1.9542; 2.3222; 3.5562; 3.0491; 3.2252. ELEMENTS OF ALGEBRA. 77 Exercise 107. 1. 4; 1; 2; -2; -1; -1; -3; -7; 0. 2. 0.8281^ 2.8281; 1.8281; T.8281; 3.8281; 6.8281. 3. "Six;" "one;" "four." 4. "Fifth;" "first." 5. T.2552; 1.3522; 0.0212; 0.5741; 1.0212; 0.7993; 2.0970; 2.6232. 6. 3.7481; 1.1070; 1.1582; 0.0970; 1.0970; 2.6990; 5.4983. 7. T.4804; 0.7323; 5.7781; 3.3222; 0.5441; 4.5441; 0.6511. Exercise 108. 1. 1.2040; 2.0970; 2.5350; 1.8060; 3.3397; 1.8060; 1.9084; 1.8572; 6.3588; 1.6110. 2. 2.5353; 2.5562; 4.1070; T5.2620 ; 3.5810; 5.2569. Exercise 109. 1. 0.3980; 1.7781; 0.1461; T.7781 ; 0..5441 ; 5.6320; T.3980; 2.4950; T.5441. 2. 0.6690; 4.1373; 0.0970; 3.7781; 1.5899; 4.1040; 0.6990 ; 0.4559. Exercise 110. 1. 0.1690; 0.1590; 0.0602; T.9466 ; 0.7952; 0.6476; T.9964 ; T.8950. 2. 0.8063 ; 0.3523 ; 1.2519 ; 0.0691 ; 0..3093 ; 0.5456. 3. T.6360; 1.6507; 0.2605; 0.6851; T.9962. 78 ANSWERS TO THE Exercise 111. 1. 1.8451; 2.0086; 2.3032; 2.9996; T.8525; 0.5563; S.8971 ; 0.5065. 2. 3.4914; 2.9926; 5.4771; 1.0034; 4.4692; 3.4794. . 3. 4.5142; 1.2638; T.5876; T.6235; 0.7672. Exercise 112. 1. 2940 ; .0289 ; 63900 ; 3.151 ; 1.57. 2. .00455 ; 30.94 ; 33138 ; .03333 ; 4.566. 3. 7.586 ; 50.56 ; 633420 ; .001301 ; 539375. Exercise 113. 1. 29.77 ; 4.75 ; 2.814 ; 32464. 2. 2.664 ; .368 ; .0769. 12. 9 ; 20 ; 121 ; 30 ; 56. 3. 373.6; .4847; .6186. 13. 2; 30; G; 14; 1. 4. 1.683; 2756; .482. 14. T.75729. 5. 9745; 3.264; 3.637; .4276; .2163. 6. 1.53+ ; 9.58+; 6.44-; 5.59+. 7. 2.56+; 1.2+; 2. 15. 2; 3; 5; 7; 10. 8. 4.56+ ; 3.46+ ; 3. 16. - 2 ; - 3 ; - 5 ; - 6 ; ^. log n log n — b log m log n 9. -3.3+ log m ' a log m ' a log m -\- b log c '09+, \ X = 1.177-f 709+ ; '1?/=: .677+ ^^ |a::::.2.709+, 1 ^ - 1.177+, 17. g ; 3 ; 4 ; 6 ; - 1. \y = 1. 18. 4; -3; 3; 6. _ 4 log m "^-"To^' (x = 3, 19. 2;|;i;-4. \ _ ^Qg^ . h = ^- 20. -y; -t;-2; 1; ^~ log 7^' -1. ELEMENTS OF ALGEBRA. 79 21. 4; 16; 8; 64; 512; 32; 256; 1024; (2048)\ 22. .00001; 5^3; 243. 23. 3; i; .3; V; V> and t?^; 4, and 4. Exercise 114. 1. ± 3; ± 2\/A- 8. ± Vm + n ; ± c. a. ±2J; ±Vi|. 9. ±\/^i ±V^f^' 3. ±8; ±5.5. 1^^ 10. ± V t; ±4. 5. ±.3; ±tV2. ''»-gl 6. ±2; ±^. "• ±"i±v!! *• *^- 7. ±|V30; ±}V77. 12. ±\/*(6»c-2«;; ± J. " c Exercise 115. 1. 12,-2; 8, -10; 16, 2. 7. ff, - y ; 0, 4. 2. 17,-4; 4,-13. 8. 2,-4,-5; 1,-1,-^. 3. -3, -i; i, -3. 9. 2, -V^; |a, ^a. 4. 107,-106; §,-i. 10 '',-^; 4,-1. 5. ^,-2; 2a, -8. , o 11- 1.-^5 ±5, ±3\/2. 6. 0,-3; T^, J. 'a -6' Exercise 116. 1. 8, 15; 7, 6; \,-\. 5. -?, -i; V-, -^3^- 2. 23,-1; 6, -J/; 1,-4. 6. V, -§; A.-?- 3. 2,-V; 3,1; 3,-1. 7. 1,-3; i,-J; ^^-J. 4. 11, V ; h -4; f -A. 8. V, -^\ ^i -f 80 ANSWERS TO THE 15. i(5±V22:6); 8,-^. 16. 11, 2; 7, 2. 17. 5, -4f. 18. 9,-H; 3,-3§§. 19. 13, ±VM; 12,-2. Exercise 117. 5. 71)— p\ ± ^2n — ^/n. / — ;- ha 1 2. ±yah\ -, — T- 6. a ± -; ^, — a. a a ^ . "^h b 2b ^ . a 3. 1, - — -; -, Q, Za, —a, a, —-' a — b a a 2 a^ __2y 8a a 5 =b V25 — 4:m^ a(a + ^) a (a — b) 9. 3,-t; -i,-6§. 10. 1,-14; ±9. 11. 13, §; 9,-ff 12. 3, -8.7; 6, e. 13. 2, ^Y-; 8± V601. 14. 12,-2; 3,-i. 1. , « b 7. (a^± ^»)2; a + b 9. (a + ^)^ - {a - by; 0, ^ 4- - - c - 1. Til 10 ^ + ^ <^-^ . a ± Va^ + 6'^ a — b' a + b' Sa^b^ a f ^b^ - 4 c'' \ l — 8a± 2aVa ^^' 2 V "^ 2«^ + ^ ; ' (a - ly 13. ± 1, ± ^ V^3; ± V2, ±^V6. ELEMENTS OF ALGEBRA. 81 Exercise 118. 1. 9H,-11; 2,4. 4. 2, i; 3,-2. 2. 7,-7|; -i,-?^ 5. i,-i; 3, f. 3. 4, -3S; 2,-8. 6. 4a, -^ a; 2,-22. 8. |(-1±V^); ^,-a,^(-5±Vr4); ±V;^-1. a -\- d ac + bm^ »• 1' — d-' "' ^ — 10. — , ^; m — 2 a, ^ m -\- a. n m, 11. a ± (6 — c). 16. 2 rt + 6 *, a — 8 ft. 2 * w — 2n' m ■\- 2n' ~^^\ -1 ± ' a + ft ± a/2 a^ + 2 ft* 13. ± V2 a - a^ -1 ± A/T^=^; - (2 ± Va^ + 4) a 14. 4,/ _ 5ft a -2ft 2 /J7 15. ^ . , » . ' 18. ±^V3. 6aft 3aft 3a Exercise 119. 2. 8, 9. 3. 15, 12. 4. 6, 21. 5. 8, 6. 7. ,»iO±\/^)- ^1^- 82 ANSWERS TO THE 9. Greater, _-«y»L^; less, _^V" _ . g. g. 10. 18. 13. 87. 14. 53, 35. 11. Eate, 30 miles an hour; 7. 12. Kate, 6 miles an hour. 2m — n ±. ^/ii^ + 4 m? 16. La]'ger pipe, ^^ minutes ; smaller, 2m^r n± V^ + 4 m^ • ^ t • oo • minutes. Larger pipe, 88 mm- utes; smaller, 154 minutes. 17. B, 6; A, 10. 18. 9 miles an hour. 19. miles an hour. 4 miles an hour. 14: am „ -a± V^^ + ^ 20. Length, ^— ^^^^' weight, /, lbs. Length, 8 feet; ^ A arn, „ .,,.-,, — a±\/ \-a^ weight, 4^ lbs. 21. <^VnTmVc Vn(VnTVc) ^^^^^^^. ^Jn^ \/c a-m Vc(Vn^:Vc) ^^^^^^^ -^g ^^yg . j^ ^1 25 ; B, $1 , a — m 22. 10 (— 5 ± Vfn + 25) dollars. 23. 10(5 ± V25 — m) dollars. Exercise 120. 1. ±2, -fcVlO; 1,-2; 2, -1, -1± ^=3, ^(1±a/^). 2. ~3; ± a, ±5. 4. 16, ifffi-; 8, W- 3. ±4, ± i; 9, (-41)1 5. ± 8, ±(-11)^; 9, ^/i681. ELEMENTS OF ALGEBKA. t 6. 3», (-28)1; (J)«, 1. 9 i JL .>_,., , tt- 4 a- * ^ ^ '■ '«.li 27. , 1 8. jm;«)u. -1,(196)". Venn (j^ 10. ±^^/2^, ± \^^; ± vS, ± v'-S. Query. Wliy, iu 10, the ± sign ? Exercise 121. 1. 2, - 3, -1 ^ 3V5 . 9. i(3 ± V5), i(9 ± ^f^^\ 2 5± V37 5± V7 2. -1, -1±2V15. "' 3 ' 3 ' 3. ± 4 V6; 18^, 5. ^^' -' ^' _ - 3 ± V33 4. 3, -i, 4 C5 ± V1329). 13. 1^ , -1,-2. 5. 5,-6, K-l± V377). 14- 1, 2, -5, 8. ®- ' ^' 3 16. 5, - 1, 2 ± V5. 7. 1,-3±2V2. -3±a/29 -3± \/I^ — ^'^' 4 ' 4 8. 4, 1,^^^. 18. 2^e^, ^rs. 10. 3, —1. 19. ^-^^, -1± V2; 3,-5, , -2 (-13 ± V313). Exercise 122. 1. 10,-2; 7, 14. 4. 14, 2.48; V- 2. 9, -12. 5. 4, -21; 5, \. 3. i, V; 8. -I- «• ^' 4' -^5 12, 4. 84 ANSWERS TO THE 7 _|_ 9 V2; 3, ^. Query. Is ^ a true value for x in the second equation ? Why ? 8. 0, ± V3. 12 i^ ^- 4 B4 2m^/n ^ ^ 9. 2^T. §; ± ^^ _^ ^ • 13. ± (^; 9, V- 10. 0,4/7; ±Vl±h^2. 14. 1, 4; 9, 4, ~ ^ ^ " ^ ' 11. ±*-'(^')'; 25, -V- 16, 0, 2,-3; ± ^^ V3. _ 7±Vl3 -lq=V^^ ,, , o-^o./^ .u. 2" ' 2 ) -^'^j -^j ^ -r- ^ V t , Exercise 123. 1. -8,-9; W, 49. 2. . J-;;, mn m — n^ 3. 0, ^; a, a^ 15, -3. 2 b, y'-a'. 5. a;2_4a.^21; Gcc^ + Sx^G; a;2-2a; = 15; x' = -Z\ 6. a;2 + 5a: = 0; x''-Ux = —29', x^ — 2 x = 1. 7. ic'-^ — (1 + m^) X = m {1 — rn^) ; mnx^ — (m^ — ?i^) a; = m w ; (a — Z*) ic^ — 4 tt 6 ic = (ct — ^) (a + hf-. 8. ic^ + 2 m a; = 8 /^ — m- ; 4 ic'^ — 4 a ic = Z» — a""^; Vox^ — '6\/'(ix^h — a. Exercise 124. 1. Imaginary ; real and equal ; imaginary. 2. Eeal and rational, and different; imaginary. 3. Real and rational, and different; real and surds, and different. KLEMKNTS OF ALGEBRA 85 4. Real and equal ; imaginary. 5. Imaginary ; real and different, and surds. 6. Real and equal; real and rational, and different; real and rational, and different. 7. Real and rational, and different, real and rational, and different ; real and surds, and different. 8. Real and rational, and different; real and surds, and different. 9. Im;i<^nnary; real and rational, and different. Exercise 125. 1. ±8, ±(-ll)i; h -i; h 4". 2. 1^, T^; -1, -ih' 3. I 450; 4, f ^"/ l ± \/l3 ^/ l ± V- 7 J ± Ks ± 2 \/4 g + 1 V 2 ' ^ 2 ' ' 2 5. ± 3, —- ; 5, -4, 6. ± h h h 7. 4, 1, 3, 2. 9. ± 2 \/2, ± \ a/=^, ± i V^185 ± 29 Vil. „ n 3±4V3 , , "• 4m», to'; -8,-4. X2. 0, -^— , 1, 4. ^^ ^_ J 13. 4, 1; J, y. 16. 0, 1, 3. 86 ANSWERS TO THE 17. -a±-— ^=; 4, 3^3. 18. (i.) a/^^ — 4 ^ a surd ; (ii.) ^^ _ 4 ^ positive; (iii.) ^2 _ 4^ negative; (iv.) yl^-4j5 = 0; (v.) J5 positive; (vi.) B negative; (yii.) B = --. j^ .'±V4j^+a- -|,^g Exercise 126. 12/ = 5, 4. ■ j2/ = 9,- 2. r.x = S, -1, J a; = 7, -5, l2/=i,-2. ■1j, = 5,-7. .^ = 6,-4, 11 \x = %-5, '• ly = 3,-7. ly = 5,-9. p-= 6,^^,-41, p = -7.4,1, *-J3, = 3i,-4. 1^ = 7.8,5. . rx = 18, 12J, <x = ll, -8, =-iy = 3,-2i. "•1y = 8,-ll ^•iy = 6,3. "•1^ = 1. p = 4,3, ^^ |. = 5,4, |y = 3, 4. l.y = 4, 5. (x = 8,17§, ig fa; = ll,-7, °- l2, = 6,-13i. l3/ = 7.-ll. Query. In 14, has x and y two values? Why.' 1. ELEMENTS OF ALGEBRA. 87 Exercise 127. (x = ±5,±7y |a: = 3, -3, 2,-2, U=±7, ±5. • -^2/ = 2, -2,3,-3. (X= i, 7, 4, f X = 5, — 5, 1, — , 1, 7. * j y = 1, - 1, 5, — o, p = 3,2, (x=±5, ±3, ^- U = 2,3. ''• |y=±3,±5. J a: = 5, -5, 3,-7, j x = 10, 4. *• 1^ = 3,-7,5,-5. • 1//-4,10. ra; = 6, 4, ^ ,3.= ± 2, ±3, ' (y = 4,6. • |y=± 3, ±2. p = 6,-6,5,-5, (0^ = 5,3, ^- |y = 5,-5,6,-6. ^^- (y = 3,5. Exercise 128. r x = ± 2, ± 1, • ly=±l, ±2. rx=±2, <x = 1. {" ±3, ±2. 2. !;: ±6, ± 2. 3. i;: ±3. 4. \l' :±2, : ±3. 5. ■■±h ■■±h ±i. ±3- 7. ± 7, ± >v/3,_ ±2, TSV3. ( ar = ± 3, ± 36, • -j y =: ± 5, :f ¥• r X = ± 8, 1 y = =F 5, ^^^x=±8,±3. 88 ANSWERS TO THE 1. r 2. 3. 5. r (y 6. f 7. f 3,5, o, 3. 6,-3, 3, -6. 8,-7, 7,-8. 10, 12, 12, 10. 10. 11. 12. 13. Exercise 129. 14.]^ \y 15. (2/ = y 16. < 5, -3,l±iV-88, -3, 5, 1 T iV--88. 3,2, 2,3. , o -13 ±V377 4, ^, ■ -^^^ , {y = ± 7, ± 5, ± 5, ± 7. a: 2, 17. 3. y 20, 15. 0, 5, 1, 18. . (0! 0,1,-1. 19. . P ±10, liS/ ±4. 20. . ^Vf 21. < a; T 1, T 2 a/=3, 2/ ± i, ± V- -1. 0,4, 0,5. 22. . [y 0,-1, 0,-2f. 23. . y y = 2, 4, = 4,3, = 3,4, 6 -13:Fi /377 6 7± V- 295 2 7:f V- 295 2, -2J, 4, -4|. 6, -2, 2, -6. 2, h 2, 16. 5 IF V15 2 5± Vi5 ±iv/l8±2V-li i(l±V-15). 1,2, 3,1. ELEMENTS OF ALGEBRA. 89 26 27 f 1 -\- ab± V(« + 1) (fl^ - 1 ) (b + 1) (6 - 1) ^*' j «& - 1 ± V(a + 1) (a - 1) (b -\-l)(b- 1) 1^^" a-b rx=±9, ±3, (:r = 2, i, -*^- -iy=:±3, ±9. • -1^ = 3,-24. ( X = ± 5, ± 2, ± 2 v^, ± 6 V^, 1 2/ = ± 2, ± 5, ± 5 V- 1, ± 2 V-H. (x = ±l, ±3, jx = 3, 2, -3± v^, (y= ±3,±1. • 1y = 2,3,-3T V3. „ ( « = ± 2, ± 3, ^ a: = 7, 1, 4 ± 2 V7, ''"• 1y=±3,±2. • ■iy:=:l,7,4:f 2V7. 30 -^ ■[y = °a^V3),j(i.-L). rx = 2,4,3q:v'21, ( a: = 5, 4, ^^- |y = 4,2, SiVai. ■<y = 4,5. 32. r=o'"!' •«>r==*^!'*l' (y = 2,-4. <y = ±4, ±9. ^^Tft' ■1y = l,9. 33. 35. „=±?1±*!. „ 5a; = 62'S,l, ^ * a-4 "• (y = 1,625. x= ±4, itV^^eB, -va6±V34lo), 90 ANSWERS TO THE ^x=± (a-h), ^x = h-h ^ =1,5. 50. 2/= ±^' ly = ^, 46. .^2/ -^,5,-,, 47. j^ = ^^' 52. 9 and 7. Query. In 47, how many values has x and y ? Why ? 63. ± }^V^((^ + ^) and ± ^ V2 (tt - 6). 11 and 7. 54. 36. a + ^-3a^ T 2 V2 (a"" + bj 55. __zi_I Z_ r — V — 2_y ^^^ Z a + ^- 3 ^2 + 2 V2 (a^ + Z>) Q ^ 1 1 1 Q — - — ^^ . 3 and 1, or 1 and 3- Z 56. — a and — 2 a, or 2 a and a. 57. 5 and 2, or — 2 and — 5, or ^ (3 ± V— 67) and i (- 3 + V=^67). 58. 6.4, or 4.6. 59. 3, 15, and 20. 2 c" • , — a ± V4 c^ + a^ 61. =:^= and . 8 and 4, or . — a ± \/4: (§ + a^ 2 — 4 and -8. 62. f , or -^f. 60. 4 and 13. 63. Time, 7 years, or 6 years ; rate, .06, or .07. 64. Principal, $10400; rate, .05. ELEMENTS OF ALGEBRA. 91 Exercise 130. ^ (x =11,8,5,2; x = 2,7,12,17, ....; a; = 8, 16, 24, 32, . .. ^' 1^ = 1,3,5,7; ^ = 7,21,35,49,....; i/ =5,8, 11, 14,.... ^ a; = 42, 31, 20, 9; x = 215, 202, 189, 176, . . . . , 7. ^ -^y = 4, 9, 14, 19; y = 5, 14, 23, 32, ...., 149. X = 8, 25, 42, 59, ,.. .; x = 7, 16, 25, 34, ....; x = 4, 17, 30, 43, .... y = 7, 22, 37, 52, ....; y = 10, 23, 36, 49, ....; y = 2, 11,20,29,.... ^x = 6;x = 9;x = 0. ^x = 10, |y = 3; 2/ = 3; y = 15. ' \y = 5. IX = 3; x = 59; x = 476. ^ x = 4, ly = '2;y= 1;3,= 19. " ^ = 24. 6. r X = 11 ; X = 37. jy=18;2/=13. Exercise 131 11. 2. 59. 6. \ and I 3. 13 and 1, or 4 and 8. 7. 131. 4. 72 and 70. 9. Nine ways. 5. ( 5 foot rod ; 19, 12, and 5. 11. No. foot rod ; 4, 9, and 14. 10. Two ways in each. 39 and 6. 12. A gives B thirteen 50-cent pieces, and B gives A forty- five 3^ cent pieces. r Horses, 1,3, 5, 7,.... (Sheep, 8,23,38,53,.... 92 • ANSWERS TO THE rl6, 15, 14, 13, ....,at^25. 14. J 2, 5, 8,11,...., at $15. 157, 55, 53, 51, ...., at $10. Sixteen ways. r 6, 13, 20, 27, 34, at 30 cents. 15. \ 45, 35, 25, 15, 5, at 45 cents. 1 24, 27, 30, 33, 36, at 80 cents. Five ways. (-74, 73, 72, ...., at 20 cents. . 16. ^4, 8, 12, .... , at 35 cents. I 72, 69, 66, ...., at 40 cents. Twenty-four ways. r23, 16, 9, 2, at $1.50. 17. \ 13, 16, 19, 22, at $1.90. l 4, 8, 12, 16, at $1.20. Four ways. 18. Four ways. 21. 269. 19. 4 at $19, 8 at $7, and 8 at $6. 22. The first has 63 or 23, the second 37 or 77. 21, 23, 25, 27, poles 7 feet long. 23. ^ 18, 13, 8, 3, poles 10 feet long. 1, 4, 7, 10, poles 12 feet long. Four ways. Exercise 132. 1. a;>i|; a^ < 4^- 12 1"^^^' 2. 3. x>-^\ X <b- x>l. -1. 13. f>^^ 9. x = 6. ly < a-b. 10. X = 4:. 15. 76 or 77. 11. x = 5. 16. 60 cents. ELEMENTS OF ALGEBRA. 93 Exercise 133. 8.i^2. a -\- b 2ab m n ^ 1 .1 5. '^±y>t^. 3(l + a'^ + a^)>(l + a + a«)^ X — y X* — y^ ^ ' 6. If«>/;, a» + 26«>3(^6^ :^ > :^; V2 4-V7 _ v6 v3 > V3 + V5. Exercise 134. 5. 4x^-3x4-1 > 6. 18 j2^<^' 6. 4-2a-'+;i2>3^^^ ' < x > 0. 7. 3 + A>a^ + 3:c«. 19- j^Jj^^^' 8. 3(a:-y)^>3(ci4-<^r; 1^ (x^ + y) > a;« - y» 10. lln'»>a2 + 8 6. ' \x >V- 11. w^ < 54; „ c < 7i«-^2^ ^ a; < _ ^, 12. -8>-27; 15>8;5>3. 13. -3>-4; 4m2 + 1>7i. 24. 2. 14. w > w ; 2 > — 4. 25. 6, or 7. 15. 2aj>m-7i; -««>/. 26. 16, or 17. 16. x< 2.9; X <3i. 27. 19. 17. jc<|; ar<i. 37. 13. 22. x> 5, ar<^; a;>-i, x<-§. 23. a; > §, X < i ; x > 7, x > - 3. 28. 2, 1, 0, -1, -2, -3, -4; -8. Vm ^ Vn miles. 126 miles. 94 ANSWERS TO THE Exercise 135. 1. -49. 5. SO a — 79 b. 9. 12. 2. 161, 245. 6. -1,0. 10. 19. 3. 16,9. 7. i. 11. -95. 4. 98, 243.6. 8. 103. Exercise 136. 12. 2, 21, and 2^. 1. 779. 4. If (9-.). 7. 76 a 4- 57 h. 2. - 5569|. 3.--^ 8. 8^. 3. a^{4:-a). 6. a7i(?z+l) 9. 13. 10. (1) 71 = 16, d = — l. (2) 7i = 7, d = 2 a. 11. 5^, 6^, 7t, etc. 13. 7500. 14. -5,-1,3,7,11. Exercise 137. 1- -^H'-6A, ••••, -^T-V 5. x''+l-x,x^+2-2x,....,x. 7n -4- 7i 2. 6.4, 5.6, ....,-5.6. 6. , , . 3. 4 ?w- — 5 ?z, 3 m — 4 w, — , 6 7i — 5 m. 4. _2l,-3§, -4f, -5|. 7. -101,-7^. 1. 48; 384. 8. ^^W, Hi W, ••.. Exercise 138. 3- — ^t; — 2T^T- 4. 128; 1. 6. x^»'- ,-288. 32^ ^' 243' « (§)'• ELEMENTS OF ALGEBUA. 95 Exercise 139. 1. 2|f 4. V(3+V3). 7. .'5. 520. 2. mh' 5. H3'"-l). 8. 2. 3. mh 6. §(1-4'"). 9. 6. 5, 30, 180. Exercise 140. 1. 42; aH^', J V6; H- 5. 40, 16, 6§. 2. ik; 6x2 — ox — 6. g _7^ ^, _|, 1^ _^, ^^. 3. 20, 80. 8. Arithmetical, f . 4. i, i. 9. 2 and 8. 7. 6, 18, 54, 162, 486, 1458, 4374. Exercise 141. 1. -4. 11. 1,9. 2. ^. 12. 20,5. 3. — ij't, — 5V' — ^> — ' iV« ^^f) — 4. 4, 2,$, I, f .... 13. 10,12,15. 5. If 14. i,i,i, 01-^,1, -J. 6 ''4,6. ^_,_^ 2a6 15. —^, Vab, 8. f , 5' A, h W, A. !«• 2550 yards. 9. 6, 12. 17. 3, 6, 12, 24. 10. 6, 2. 96 ANSWERS TO THE ELEMENTS OF ALGEBRA. Exercise 142. 1. 54 6 : a. 10. -13. 2. 9:7. 11. 2, or 3. 3. 10 : 9. 12. 7: 2. 4. (l-y)(l + x):l+x\ 13. an — hm 5. 14. m — n h in — an 6. 5a:4; Ay : 6x\ Sy : X. m 7. 28 ic : By, n — 1 : 2a. 15. 4:7. 8. 7 : 8, 31 : 36, 41 : 48, 5 :6. 16. "^ah. 9. a" — h^-.a"- -\-h'- >a — h\a^-h. Exercise 143. 3. 4; 2; t33-V210; 12; ^2^2^ 29. h '. a. 4. ^2 _ ^2 . 300 ^8 j^^ g^ ^^ 5- M; 3r\; .8; 4^; 1^. 3-,^ 4 . 1^ a?" 6. 3"; (a + ^)2; ^f--,-.- 32. 17:7. 7. 15; i; 2^3^; 6^-JZ». ^^- '"^ * ^• 8. 5/;2; 1. 34. 5: 4. 21. a = c' c?3, ^ = c^ (^i 35. V&xy. 23. a.= t^^a^^;a:-3,or-l. 37. ^ = ^' ■ ^^ ' b n^ 24. ( a; z!r i 4, i- 6, ^^a + bV „ . ^ ^ / 39. £c = ^; — ^7 ); ir = 3,or— 1. (y = ±6, ±4. W^y 27. A invested $3000 ; B, $3500. 38. Bate of slow train ; rate of fast train : : 1 : 2. or THE ^ ^ f UMIVERSITY I \ K^ ^^ :/ A *;: - (■ THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO 50 CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. SEf 20 \^ SEP 1* t»^ n = 17Nov'57HIVI peco ^ «0M 8*^ / y a a